More Related Content Similar to Algebra de funciones (20) More from Widmar Aguilar Gonzalez (20) Algebra de funciones 1. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
EJERCICIOS RESUELTOS DE MATEMATICA
BASICA
Temas:
- AXIOMAS DE ORDEN
- DOMINIO DE FUNCIONES
- ALGEBRA DE FUNCIONES
- COMPOSICION DE FUNCIONES
- FUNCIONES: INYECTIVAS
- FUNCIONES: INVERSAS
Ing. WIDMAR AGUILAR, Msc
julio 2021
21. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
1)
2 β 3 +6=0 β = 2 + 6
Encontremos dos puntos y trazar la recta que pase por ellos
X= 0 β = 2 ---------A(0,2)
= 3 β = 4 β β β β 3,4
De la grΓ‘fica se puede determinar dominio y rango de la funciΓ³n:
D(f) = R
Ran (f) = R
2)
β 2 + β 1 = 0
+ 1 =1+2x
= ; β β1
22. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
Realizando su grΓ‘fica, se puede observar el dominio y
rango de la funciΓ³n:
D (f) = R-{-1}
De: β 2 + β 1 = 0
β 2 = 1 β
= ; y β 2
Ran (f) = R β{2}
3)
Del dato se define:
Ran (f) = ]2, 6]
3y= 2x+8
23. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
= 2 + 8
Si: = 2 β = β1 ; β1,2 ββ !
= 6 β = 5 ; 5,6
La grΓ‘fica de la funciΓ³n es una recta:
D(f) = ]-1, 5]
4)
4 + 4 β 16 + 4 β 47 = 0
4 β 16 + 4 + 4 β 47 = 0
4 β 16 + 16 + 4 + 4 + 1 β 47 β 17 = 0
4 β 4 + 4 + 4 $ + +
%
& = 64
4 β 2 + 4 $ + & = 64
β 2 + $ + & = 16 β β β 'Γ)'*+,
24. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
C(h,k) = C(2, -1/2) ; r = 4
- ! = .β β ), β + )0 = .β2,60
123 ! = .4 β ), 4 + )0 = 5β
6
,
7
8
5)
= 1 β β15 β 2 β
= 1 β :β + 2 + 1 + 16
= 1 β :16 β + 1 ---------semicirculo (hacia abajo)
16 β + 1 = β 1
+ 1 + β 1 = 16
; β, 4 = ; β1,1 ; ) = 4
25. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
- ! = .β β ), β + )0 = .β5,30
123 ! = .4 β ), 40 = .β3,10
6)
= β3 + β4 β ---semicircunferencia
+ 3 = :β β 4 + 4 + 4
+ 3 = :4 β β 2
+ 3 = 4 β β 2
β 2 + + 3 = 4
) = 2 ; ; β, 4 = ' 2, β3
26. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
- ! = .β β ), β + )0 = .0,40
123 ! = .4, 4 + )0 = .β3, β10
7)
= 2 + :6 β
β 2 = :β β 6 + 9 + 9
β 2 = :9 β β 3
β 2 = 9 β β 3
β 2 + β 3 = 9
β = 2 + :6 β β β β β β β=>?@'@)'*3!>)>3'@2
; β, 4 = ; 2,3 ; ) = 3
- ! = .β, β + )0 = .2,50
27. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
123 ! = .4 β ), 4 + )0 = .0,60
8)
+ β 2| | β 6 + 1 = 0
| | = B
, β₯ 0
β , < 0
a) X <0 ; + + 2 β 6 + 1 = 0
+ 2 + 1 + β 6 + 9 + 1 β 1 β 9 = 0
+ 1 + β 3 = 9------circunferencia
; β, 4 = ; β1,3 ; ) = 3
b) X >0 ; + β 2 β 6 + 1 = 0
β 2 + 1 + β 6 + 9 + 1 β 1 β 9 = 0
β 1 + β 3 = 9------circunferencia
; β, 4 = ; 1,3 ; ) = 3
- ! = .β4,40
123 ! = .0,60
28. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
9)
+ 2 β 2 + 7 = 0
2 = + 2 + 7
2 = + 2 + 1 + 6
= + 1 + 3 β β β βE2)Γ‘G,+2
Que se abre hacia arriba
β = β1, 4 = 3
V(h.k) = V(-1,3)
- ! = 1
123 ! = .3, β .
10)
2 β 4 + + 3 = 0
= 4 β 2 β 3
29. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
= β 2 β 4 β 3
= β2 β 2 + 1 β 3 + 2
= β1 β 2 β 1 β β β βE2)Γ‘G,+2
V(h, k) = V(1,-1)
2 < 0 β => 2G)> β2'@2 2G2I,
- ! = 1
123 ! = 0 β β , β10
11)
+ 4 + 3 β 8 = 0
+ 4 + 4 + 3 β 8 β 4 = 0
3 = β + 2 + 12
= β4 β + 2
30. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
β β < 0 β β β β β => 2G)> β2'@2 +2 @JK*@>)L2
V(h,k )= V(-2, -4)
- ! = 0 β β , 40
123 ! = 1
12)
= 1 + β2 β
De: y= k+:β β β β β β =>?@E2)Γ‘G,+2
Que se abre hacia la izquierda.
= 1 + :β β 2
h = 2 ; k=1
31. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
- ! = 0 β β , 20
123 ! = .1, β .
13)
= ββ6 β 2
= β :β2 β 3 = ββ2 :β β 3
Se sabe: y = 4 β G:β β β β β=>?@E2)Γ‘G,+2
-β2 < 0 β => 2G)> β2'@2 +2 @JK*@>)L2
= ββ2 :β β 3
h = 3 ; k = 0
- ! = 0 β β , 30
123 ! = 0 β β, 0 0
14)
32. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
+ 1 = β3 + 5
= β1 + M3 +
N
= β1 + β3 M +
N
Si: y = k+b β β β β β β β=>?@E2)Γ‘G,+2
Que se abre a la derecha
h = -5/3 ; k = -1
V(-5/3, -1)
- ! = .β
N
, β .
123 ! = .β1, β.
15)
= 5 + :β3 β 2
= 5 + β3 :β β 2
-----semiparΓ‘bola que se abre a la izquierda
h = 2 ; k =5
V (2,5)
33. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
- ! = 0 β β, 20
123 ! = .5, β .
16)
4 + 9 β 16 + 18 = 11
4 β 4 + 9 + 2 = 11
4 β 4 + 4 + 9 + 2 + 1 = 11 + 25
4 β 4 + 4 + 9 + 2 + 1 = 36
O
6
+
O
%
= 1 β β β >+@E=>
B
2 = 3
G = 2
; β, 4 = ; 2, β1
- ! = .β β 2, β + 20 = .β1,50
34. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
)23 ! = .4 β G, 4 + G0 = .β3, 10
17)
9 + 4 + 18 β 32 = β37
9 + 2 + 1 + 4 β 8 + 16 = β37 + 73
9 + 1 + 4 β 4 = 36
O
%
+
% O
6
= 1 β β β β β >+@E=>
2 = 2 ; G = 3
C(h, k) = C (-1, 4)
La grΓ‘fica es:
- ! = .β β 2, β + 20 = .β3,10
123 ! = 4 β G, 4 + G0 = 1,70
18)
35. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
= + | β 1|
| β 1| = P
β 1 ; β₯ 1
β β 1 ; < 1
a) X < 1 ; = β + 1 = 1
b) X β₯ 1 = + β 1 = 2 β 1
! = B
2 β 1 ; β₯ 1
1 ; < 1
La grΓ‘fica es:
- ! = 1
123 ! = .1, β .
19)
=
| |
+
36. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
| β 1| = P
β 1 ; β₯ 1
β β 1 ; < 1
a) X < 1 ; = +
= 1 +
b) X β₯ 1 = + = = β +
= β 1
! = B
β 1 ; β₯ 1
1 + ; < 1
Su grΓ‘fica es:
- ! = 1 β 1Q
123 ! = 1
20)
| | + | | = 4
| | = 4 β | |
37. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
a) X β₯ 0 ; | | =
| | = 4 β β 4 β β₯ 0 β§ 4 β = Γ³ 4 β = β Q
X β₯ 0 β β€ 4 β§ = 4 β Γ³ = β 4 Q
X β₯ 0 β§ β€ 4 β 0 β€ β€ 4 β = 4 β Γ³ = β 4Q
b) X <0 ; | | = β
| | = 4 + β 4 + β₯ 0 β§ 4 + = Γ³ 4 + = β Q
X < 0 β§ β₯ β4 β β4 β€ < 0 β = 4 + Γ³ =
β β 4 Q
β4 β€ β€ 0 β = 4 + Γ³ = β β 4Q
0 β€ β€ 4 β ! = B
4 β
β 4
β4 β€ β€ 0 β ! = B
4 +
β β 4
V2 W)Γ‘!@'2 >=:
- ! = .β4,40
123 ! = .β4,40
38. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
21)
| + 2| + | β 3| = 4
| β 3| = 4 β | + 2|
| + 2| = P
+ 2 ; β₯ β2
β + 2 ; < 2
a)
β₯ β2 β | + 2| = + 2 β | β 3| = 2 β β 2 β β₯ 0 β§
{ y-3=2-x Γ³ y-3 = x-2 }
β₯ β2 β§ β€ 2 β β2 β€ β€ 2 β β2 β€ β€ 2 β§ =
5 β Γ³ = + 1Q
b)
< β2 β | + 2| = β β 2 β | β 3| = 6 + β 6 + β₯ 0
β§ { y-3=6+x Γ³ y-3 = -x-6 }
< β2 β§ β€ β6 β β6 β€ β€ β2 β β6 β€ β€ β2 β§
= 9 + Γ³ = β3 β Q
-2β€ β€ 2 β ! = B
5 β
+ 1
β6 β€ β€ β2 β ! = B
9 +
β β 3
V2 W)Γ‘!@'2 >=:
39. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
- ! = .β6, 20
123 ! = .β1, 70
22)
= | + 4 + 1|
= | + 4 + 4 β 3|
= | + 2 β 3|
β β₯ 0 β§ = + 2 β 3 Γ³ = β + 2 + 3 Q
De:
= + 2 β 3 β β β β β E2)Γ‘G,+2
β = β2 ; 4 = β3 ; Y β2,3
= β + 2 + 3 β β β β β E2)Γ‘G,+2
β = β2 ; 4 = 3 ; Y β2,3
40. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
- ! = 1
123 ! = .0, β .
23)
! + 2 =
Con x = 1 β ! 3 = 1 β 3,1 β !
Con x =-1 β ! 1 = β1 β 1,1 β !
De; (x,y ) β ! β§ , J β ! β = J
3,1 β§ 1, β1 β !
Luego --------------- f es una funciΓ³n
24)
41. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
! + 2 =
Con x = 1 β ! 3 = 1 β 3,1 β !
Con x =-1 β ! 3 = β1 β 3, β1 β !
De; (x,y ) β ! β§ , J β ! β = J
3,1 β§ 3, β1 β ! 1 β β1
Luego --------------- f no es una funciΓ³n
25)
Sea t = x+3 ; x = t-3
! [ = ([ β 3 β 1
! [ = [ β 6[ + 9 β 1 = [ β 6[ + 8
β ! = β 6 + 8
Calculando:
]
]
=
] O ^ ] _
]
=
]O %] % ^] _ % _
]
=
]O ]
]
=
] ]
]
= 2
]
]
= 2
26)
42. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
Calculando f(x).
Sea: x+1 = t
! [ = [ β 1 + 3 = [ β 2[ + 1 + 3
= [ β 2[ + 4
! = β 2 + 4
] ]
]
=
] O ] % . ] O ] %0
]
=
]O %] % ] % % ]O %] % ] % %
]
=
_] _
]
=
_ ]
]
= 8
] ]
]
= 8
27)
La grΓ‘fica es:
43. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
Sea: 0<x β€1
1< <
`
β 1 < 1/
> 1
1 + > 2
! > 2
β ! β 01,2. β β β β β!2+=,
28)
= β 1
Su grΓ‘fica es:
44. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
- ! = β
Si:
X= 1 β ! 1 = 0 β 1,0 β !
= β1 β ! β1 = 0 β β1,0 β !
(x,y ) β ! β§ , J β ! β = J
1,0 β ! β§ β1,0 β !
! >= *32 !*3'@Γ³3
Despejando x:
= + 1
= Β± : + 1
+ 1 β₯ 0
β₯ β1
Rang(f) = [1, β .
29)
45. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
β 4 β 2 + 10 = 0
2 = β 4 + 10
= β 4 + 10
Como es polinomio β - ! = β
= β 4 + 4 + 5 β 2
= β 2 + 3 β E2)Γ‘G,+2
Y β, 4 = 2, 3
Si x =1 β ! 1 =
7
β $1,
7
& β !
= β1 β ! β1 =
N
β $β1,
N
& β !
$1,
7
& β ! β§ $β1,
N
& β !
f es una funciΓ³n
2( y-3) = ( β 2
:2 β 6 = + 2
= :2 β 6 + 2
Escriba aquΓ la ecuaciΓ³n.
2 β 6 β₯ 0 β β 3 β₯ 0
β₯ 3
Luego el rango es: Rang(f) = [3, β .
46. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
30)
= 3 + 2 β
= β β 2 + 1 + 3 + 1
= 4 β β 1
Si: β2 β€ < 2
-3 β€ β 1 < 1
0β€ β 1 < 9
β9 < β β 1 β€ 0
4 β 9 < 4 β β 1 β€ 4
β5 < ! β€ 4
Rang (f) = ]-5, 4 ]
47. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
31)
= 1 + β3 + 2 β
= 1 + :β β 2 + 1 + 4
= 1 + :4 β β 1
Partiendo del dominio: β1 < β€ 2
-2 < x-1 β€ 1
0 β€ β 1 < 4
β4 < β β 1 β€ 0
0 < 4 β β 1 β€ 4
0 < :4 β β 1 β€ 2
1 < 1+ :4 β β 1 β€ 3
Rang (f)= ]1, 3 ]
48. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
32)
Se traza la cuadrΓcula con los datos del dominio y rango y en Γ©l la
curva f(x)
Y= 0 β 0 = + 4 + 4 β 1
0 = + 2 β 1 β + 2 = 1
β = β2 Β± 1
β P
= β3
= β1
49. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
= 5 β 5 = + 4 + 4 β 1
5 = + 2 β 1 β + 2 = 6
β = β2 Β± β6
β r
= β2 β β6
= β2 + β6
β β4 β β 16 β 16 + 3 = 3
(-4, 3) β ) !
β 1 β β 1 β 4 + 3 = 0
(1, 0) β ) !
Luego:
D= D(g) = ]-4, 3[ U ]-1, β6 β 2.
Rang(g) = f(D) = [0, 5]
33)
Se traza la cuadrΓcula con los datos del dominio y rango y en Γ©l la
curva f(x)
50. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
Si:
= 3 β 3 = β9 β
9 = 9 β
x= 0
= 1 β 1 = β9 β
1 = 9 β
= 8 β = Β±2β2
= 2 β = β9 β 4
y = β5
Luego:
D= D(g) = [-2, 2]
Rang(g) = f(D) = [β5, 3]
34)
Se traza la cuadrΓcula con los datos del dominio y rango y en Γ©l la
curva f(x)
51. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
Y= 7 β 7 = 3 β 2
2 = β4 β = β2
Y= -1 β β1 = 3 β 2
2 = 4 β = 2
D= D(g) = [-2,2[
Rang(g) = f(D) = ]-1,7]
35)
Se traza la cuadrΓcula con los datos del dominio y rango y en Γ©l la
curva f(x)
52. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
A= [-2,3[
B= [-1,2]
f (x) = x2 -2
W: - β / ! = W
Si;
Y= -1 β β1 = β 2
= 1 β = Β±1
Y= 2 β 2 = β 2
= 4 β = Β±2
D = D(g) = .β2, β10s .1,2]
Rang(g) =f(D) = [-1,2]
36)
53. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
A = [-2,3[ ; B = [-2, 6[
Trazar Ax B y dentro del rectΓ‘ngulo la funciΓ³n f( x)
β 6 β 6 β β 9
β 15 β β Β± β15 ---fuera de Ax B
= β2 β β2 = β 9
= 7 β = Β± β7
= β7 β β β => L>='2)[2 >+ t2+,) 3>W2[@t,
Se tiene:
D = D(g) = [β7 , 3.
Rang(g) = [-2,0[
54. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
37)
! = | | + | β 1|
De la definiciΓ³n de valor absoluto:
| | = B
, β₯ 0
β , < 0
| β 1| = P
β 1 , β₯ 1
β β 1 , < 1
D(f) = A = [-3,3]
Redefiniendo la funciΓ³n f(x):
]-3, 0[ β | | = β ; | β 1| = β β 1
! = β β β 1 = 1 β 2
[0, 1[ β | | = ; | β 1| = β β 1
! = β β 1 = 1
[1, 3] β | | = ; | β 1| = β 1
! = β 1 = 2 β 1
! = u
1 β 2 ; β 0 β 3,0.
1 ; β .0,1 .
2 β 1 ; β .1, 30
Traza A x B y dentro de este perΓmetro a funciΓ³n f(x);
55. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
Si:
Y= 2 β 2 = 2 β 1 β =
= 5 β 5 = 2 β 1
= 3
Y= 2 β 2 = 1 β 2 β = β
Y= 5 β 5 = 1 β 2 β = β2
D= D(g) = ]-2,-β s 0 , 30
Rang (g) = ]2,5]
38)
D(f) = [-2,-3[
Trazar la cuadricula A x B y dentro de ella dibujar la curva f( x):
56. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
La funciΓ³n f( x ) es una parΓ‘bola con vΓ©rtice: V(0,-9)
= β2 β β2 = β 9
β = 7 β = Β± β7
= β7
Se tiene:
D= D(f) = . :7, 30
Rang (g) = [-2,6[
39)
D(f) = ?
! = M
O
| N|
De:
O
| N|
β₯ 0
57. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
%
| N|
β₯ 0 ; |2 β 5| = u
2 β 5 , β₯
N
β 2 β 5 . <
N
a) 0 β β,
N
. β |2 β 5| = 5 β 2
f(x) =
%
N
β₯ 0 β + 3 4 β 5 β 2 β₯ 0
S1 = β .β3,
N
. s β₯ 4
De: .β β,
N
. β© β .β3,
N
. s β₯ 4
S1 = β .β3,
N
.
B ) .
N
, β . β |2 β 5| = 2 β 5
f(x) =
%
N
β₯ 0 β + 3 4 β 2 β 5 β₯ 0
58. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
De: .
N
, β . β© x β€ β3 s 0 5/2, 40
S2 = 0 5/2, 40
La soluciΓ³n serΓ‘: w1 s w2
S = [ 3,4] β{5/2}
40)
Sea: ! = 3 β β2 β
= 3 β β2 β β 2 β = 3 β
X = 2 β 3 β β = 2 β β 3
59. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
β E2)Γ‘G,+2 L> tΓ©)[@'> =
Y 2,3 => 2G)> β2'@2 +2 L>)>'β2
Se tiene que:
Rang(f) = ]- β , 30
Sea: g(x) = x2 +14x +50
W = + 14 + 49 + 1
W = + 7 + 1
β E2)Γ‘G,+2 K*> => 2G)> β2'@2 2))@G2
V(h,k) = (-7, 1)
60. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
En rango es: Ran (g) = [1, β .
Luego; Rang(f) β© 123 W
Rang(f) β© 123 W = .1, 30
41)
De: β1 < ! < 3; β β 1
β1 <
O ]
O < 3
Se tiene que: + 2 + 2 > 0 E,) =>) β < 0
β = L@=')@?@323[>
-( + 2 + 2 < 2 β 2 + 1 < 3 + 2 + 2
β -( + 2 + 2 < 2 β 2 + 1 β§ 2 β 2 + 1 < 3 +
2 + 2
a) -( + 2 + 2 < 2 β 2 + 1
β 0 < 3 + 2 β 2 + 3
β 3 + 2 β 2 + 3 > 0
Debe cumplirse que el discriminante sea menor que
cero:
β < 0
61. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
β 2 β 2 β 36 < 0 β 2 β 2 < 36
β . 2 β 2 β 60.2 β 2 + 60 < 0
β β 2 + 4 8-a) < 0
β 2 + 4 8 β 2 > 0
2 β 0 β 4, 8 .
b) 2 β 2 + 1 < 3 + 2 + 2
β 0 < + 2 + 6 + 5 > 0
Debe cumplirse que el discriminante sea menor que
cero:
β < 0
2 + 6 β 20 < 0
β 2 + 6 < 20 β β β20 < 2 + 6 < β20
β ββ20 β 6 < 2 < β20 β 6
2 β0 β β20 β 6, β20 β 6 .
Finalmente se tiene:
2 β 0 β 4, 8. β§ 2 β0 β β20 β 6, β20 + 6 .
2 β 0 β 4, β20 β 6.
62. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
42)
! > 1
^ O { `
|N O |
> 1
5 β 3 + 1 β β < ,
Como el discriminante es menor que cero, la expresiΓ³n siempre
serΓ‘ positiva.
5 β 3 + 1 > 0
β 6 + 2 ? + 10 > 5 β 3 + 1
β + 2 ? + 3 + 9 > 0 β β β β β β 2
Se tiene que (a) es positivo β β < 0
2? + 3 β 36 < 0
β 2? + 3 < 36
β β6 < 2? + 3 < 6
β β9 < 2? < 3
β β
6
< ? <
? β 0 β
6
, .
43)
Si fβ¦β¦. es cuadrΓ‘tica β ! = 2 + G + '
63. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
! $ β 1& β ! $ + 1& = β8 + 1 ------(1)
Si:
! 0 = 1 β 1 = 2 0 + G 0 + '
' = 1
! = 2 + G + 1
De la expresiΓ³n (1),
2 β 1 + G $ + 1& + 1 β .2 $ + 1 + G $ + 1& + 18 =
β8 + 1
2 $
O
%
β + 1& + G. + G + 1 β .2 $
O
%
+ + 1& + G. + G +
10 = β8 β 8
β22 β 2G = β8 β 8
2 + G = 4 + 4
β B
2 = 4
G = 4
La ecuaciΓ³n f(x) serΓ‘:
! = 4 + 4 + 1
! = 2 + 1
β
E2)Γ‘G,+2 K*> => 2G)> β2'@2 2))@G2 ',3 tΓ©)[@'> Y β, 4
Y β, 4 = β = β ; 4 = 0
64. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
De la grΓ‘fica β >+ ?Γ3@?, L> ! >=: 0
Min. f = 0
44)
Sea: ! = 2 + G + '
|
2 = 2
G = 2β5 β 1
' = ββ5
! = 2 } +
βN
+
~ βN β’
O
^
β¬ β β5 β
~ βN β’
O
_
! = 2 +
βN
%
β β5 +
~ βN β’
O
_
! = 2 +
βN
%
β
_βN ` %βN
_
! = 2 +
βN
%
β
%βN
_
β E2)Γ‘G,+2 K*> => β2G)> β2'@2 2))@G2.
β u
β = β
βN
%
4 = β
%βN
_
La grΓ‘fica de f(x) es:
65. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
El mΓnimo de f es: β
%βN
_
45)
P = + 2 +
β’
= 2
β 2 + 4 + β = 4
2 + β + 4 = 4
=
%
.4 β 2 + β 0
66. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
El Γ‘rea de la ventana es:
A= xy +
β’
=
%
.4 β 2 + β 0+
β’
_
= β
O
%
2 + β +
β’
_
= β
O
β
β’
%
+
β’
_
= β
O
β
β’
_
= β
O
β
β’
_
A(x) = β +
β’
_
β β β β E2)Γ‘G,+2
Se abre hacia abajo
A(x) = β $
% β’
_
& = β. $
% β’
_
& β 0
= β. $
% β’
_
& β + $
_
β’ %
& 0+ $
_
β’ %
&
= β. . M
% β’
_
β M
_
% β’
0 + $
_
β’ %
&
Y β, 4 = ΖM
_
% β’
, $
_
β’ %
& β
67. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
Derivando la funciΓ³n A(X) e igualando a cero para tener un
mΓ‘ximo:
β¦
= 0
0 = 1 β 2 $
% β’
_
&
2 $
% β’
_
& = 1
=
%
β’ %
46)
Del triΓ‘ngulo ;- β ;1:
β‘Λ
β°β°β°β°
Ε βΉ
β°β°β°β°
=
Εβ’
β°β°β°β°β°
ΕΕ½
β°β°β°β°
β
`
=
Εβ’
β°β°β°β°β°
^ β’
68. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
20 6 β β = 6
β = 6 β
`
El Γ‘rea del rectΓ‘ngulo es: A= x.h
= . $6 β
`
& = β
`
+ 6
= β
`
β 20
= β
`
β 20 + 100 + 30
= 30 β
`
β 10 ////
β E2)Γ‘G,+2 K*> => 2G)>2 β2'@2 2G2I, β¦
V(h,k) = ( 10, 30)
El mΓ‘ximo se tiene cuando X = 10
X= 10 β = 30
30 = . β = 10. β
β = 3
B
= 10
β = 3
69. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
47)
= β1 ; ',?,: 10 β = 2β1
1 =
`
β’
= β
`
β’
=
` O
%β’
= $%
& =
^
= + =
` O
%β’
+ ^
=
`` ` O
%β’
+
^
=
%`` _` % O β’ O
^β’
=
O
^
+
O
%β’
β
N
β’
+
N
β’
70. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
A= $ ^
+
%β’
& β
N
β’
+
N
β’
= $
β’ %
^β’
& β
N
β’
+
N
β’
=
^β’
. 4 + β β 80 +
^``
% β’
0 +
%``
^β’
β
^``
^β’ % β’
=
^β’
. β4 + β β
%`
β% β’
0 +
N
β’
β
``
%β’ β’O
β E2)Γ‘G,+2 K*> => 2G)> β2'@2 2))@G2 β β β β β
El mΓnimo de fβ¦. se tiene para x =
%`
% β’
El perΓmetro del cuadrado serΓ‘:
P1 = 4(x/4) =x
P1=
%`
% β’
La longitud de la circunferencia:
L2= 2β1 = 2β $
`
β’
&= 10 β
V2 = 10 β
%`
% β’
=
`β’
% β’
El Γ‘rea del cuadrado es:
= $
4
& =
1
16
= ^
%`
% β’
71. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
El Γ‘rea total es:
A =
` O
%β’
+
^
=
^
%`
% β’
+
`
ββ
βββ
O
%β’
A =
N
% β’
β’β
β’
=
β
βΛ
ββ
βββ
O
Oβ’
βββ
=
%
% β’
Ε‘ =
%
% β’
.
N
% β’
=
``
% β’ O
48)
De; ! =
%
=
% %
%
= 1 +
%
%
β β 4 β₯ 0
β > 0 β β₯ 0
β β₯ 0 β§ β 4 β₯ 0
β₯ 0 β§ (x+2)(x-2) β₯ 0
β₯ 0 β§ { β€ β2 Γ³ β₯ 2 Q
β .2, β . β 4Q
D (f) = .2, 4 . s 04, β .
El rango serΓ‘:
2 β€ < 4 Γ³ > 4
β2 β€ β 4 < 0 Γ³ β 4 > 0
%
β€ β Γ³
%
> 0
72. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
%
%
β€ β2 Γ³
%
%
> 0
1+
%
%
β€ β1 Γ³
%
%
> 0 + 1
1+
%
%
β€ β1 Γ³
%
%
> 1
Ran (f) = 0 β β, β1 0 s 01, β.
49)
Se tiene:
! = 4 β β + 12 + 27 ; β0 β β, β110
W = + 6 + 6 ; β 00, β .
+ 12 + 27 = + 12 + 36 + 27 β 36
= + 6 β 9
! = 4 β : + 6 β 9
W = + 6 + 9 + 6 β 9
W = + 3 β 3
Se determina el rango a partir del dominio de f.
< β11
+ 6 < β5
+ 6 β₯ 25 β + 6 β 9 β₯ 16
: + 6 β 9 β₯ 4
β: + 6 β 9 β€ β4
4 β : + 6 β 9 β€ 0
73. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
! β€ 0
Ran (f) = ] ββ, 0 0
De: x >0
+ 3 > 3
+ 3 > 9
+ 3 β 3 > 6
W > 6
123 W = 06, β .
50)
De: β1 < ! < 3; β β 1
β1 <
O βΊ
O < 3
Se tiene que: + 2 + 2 > 0 E,) =>) β < 0
β = L@=')@?@323[>
-( + 2 + 2 < 2 β 4 + 1 < 3 + 2 + 2
β -( + 2 + 2 < 2 β 4 + 1 β§ 2 β 4 + 1 < 3 +
2 + 2
a) -( + 2 + 2 < 2 β 4 + 1
β 0 < 3 + 2 β 4 + 3
β 3 + 2 β 4 + 3 > 0
Debe cumplirse que el discriminante sea menor que
cero:
74. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
β < 0
β 2 β 4 β 36 < 0 β 2 β 4 < 36
β . 2 β 4 β 60.2 β 4 + 60 < 0
β β 4 + 4 8-k) < 0
β 4 + 4 8 β 4 > 0
4 β 0 β 4, 8 .
b) 2 β 4 + 1 < 3 + 2 + 2
β 0 < + 4 + 6 + 5 > 0
Debe cumplirse que el discriminante sea menor que
cero:
β < 0
4 + 6 β 20 < 0
β 4 + 6 < 20 β β β20 < 4 + 6 < β20
β ββ20 β 6 < 4 < β20 β 6
4 β0 β β20 β 6, β20 β 6 .
Finalmente se tiene:
4 β 0 β 4, 8. β§ 4 β0 β β20 β 6, β20 + 6 .
4 β 0 β 4, β20 β 6.
75. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
51)
! = | β 1| + | + 1|
Puntos crΓticos = { 1, -1}
2 0 β β, β1. β | β 1| = β β 1
| + 1| = β + 1
! = β β 1 β + 1 = β + 1 β β 1
! = β2
b.- .β1,1. β | β 1| = β β 1
| + 1| = + 1
! = β β 1 + + 1 = β + 1 + + 1
! = 2
c.- .1, β. β | β 1| = β 1
| + 1| = + 1
! = β 1 + + 1 = β 1 + + 1
! = 2x
Redefiniendo a f:
! = |
β2 , < β1
2 , β β1,1 .
2 , β₯ 1
La grΓ‘fica es:
76. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
De la grΓ‘fica se aprecia que el rango es:
123 ! = .2, β .
52)
De: β 3 β 4 β₯ 0
β 4 + 1 β₯ 0
β€ β1 Γ³ β₯ 4
β - ! = 0 β β, β10 s .4, β .
β 3 β 4 = β 3 + 9/4 β 4 β 9/4
= β β
N
%
De:
β€ β1 Γ³ β₯ 4
β β€ β
N
Γ³ β β₯
N
β β₯
N
%
Γ³ β β₯
N
%
77. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
β β
N
%
β₯ 0 Γ³ $ β & β
N
%
β₯ 0
β β
N
%
β₯ 0
! β₯ 0
123 ! = .0, β .
Para graficar, se parte de la ecuaciΓ³n dada:
! = β β 3 β 4
! = M β β
N
%
= M β β
N
%
β = β β
N
%
+
N
%
= β
$ β & β =
N
%
$
Ε
O
&
O
Oβ’
β
β
O
Oβ’
β
= 1 β β@EΓ©)G,+2
u
2 =
N
G =
N
Pero como se tiene la raΓz cuadrada -----la mitad de la hipΓ©rbola
78. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
53)
! =
O %
; β β3
! = β 4 β 1
Como f(x) es polinomio β - ! = 1 β 3Q
! = β 4 + 4 β 5
! = β 2 β 5
La grΓ‘fica fβ¦es una parΓ‘bola que se abre hacia arriba y de vΓ©rtice
V(h,k)
Y β, 4 = 2, β5
Ran (f) = .5, β .
79. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
54)
Factorizando:
! =
O `
N
; β β1, β β5
! =
N
N
; β β1, β β5
! = β 2
- ! = 1 β β1, β5Q
= β 2 β = + 2
123 ! = 1 β β3, β7Q
80. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
55)
! = | |. | β 1|
Puntos crΓticos = {0,1}
a) 0 β β, 0. β | | = β
| β 1| = β β 1
! = β 1 = β
G .0, 1. β | | =
| β 1| = β β 1
! = β β 1 = β
' .1 , β. β | | =
| β 1| = β 1
! = β 1 = β
! = u
β , < 0
β , 0 β€ < 1
β , β₯ 1
D (f) = R
Para determinar el rango se puede realizar el grΓ‘fico:
81. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
Ran (f) = [0, β .
56)
L> +,= L2[,= L> +2 !*3'@Γ³3 2 [)2J,=, => [@>3>:
- ! = .β2, 10 s 01, 40
β2 β€ β€ 1
β4 β€ 2 β€ 2 β β3 β€ 2 + 1 β€ 2 + 1
β3 β€ ! β€ 3
Ran(f1) = [-3, 3]
!2 = β 3 = $ β 3 +
6
%
& β
6
%
! = $ β & β
6
%
De: 1< x β€ 4
82. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
- - < x - β€
N
0 β€ $ β & β€
N
%
β
6
%
β€ β 3 β
6
%
β€ 4
123 ! = 5β
6
%
, 48
Ran (f) = Ran (! + 123 !
= [-3, 3] U 5β
6
%
, 48
123 ! = .β3, 40
β E2)Γ‘G,+2 L> tΓ©)[@'> $ , β
6
%
& K*> => 2G)>
β2'@2 2))@G2
83. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
57)
Factorizando se tiene:
+ β 2 β 2 = + 1 β 2 + 1
= + 1 β 2
Ε O
=
~ O β’
= β 2 ; β β1
! = P
β 2 ; β .β3,2.β β1Q
8 β 2 ; β .2, 4 .
Sea:
! = β 2 β
E2)2G,+2 K*> => 2G)> β2'@2 2))@G2
β = 0 ; 4 = β2
La grΓ‘fica serΓ‘:
84. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
D(f) = [-3, 4 [-{-1|
Ran (f) = [-2, 7[
58)
Reescribir la funciΓ³n con valor absoluto
| + 3| = P
+ 3 ; β₯ β3
β + 3 ; < β3
a) 0 β 5, β3. β | + 3| = β β 3
b) .β3, β10 β | + 3| = + 3
! = β’
β β 3 ; β0 β 5, β3.
+ 3 ; β . β3, β1 0
2 ; β 0 β 1, 20
12 β 2 ; < 2
- ! = .β5, β .
La grΓ‘fica es:
85. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
123 ! = 0 β β , 8 0
59)
Sea: ! = β β 9
= β 9 β β = 9
O
6
β
O
6
= 1 β β@EΓ©)G,+2
! = β β 9 ; =>?@ β@EΓ©)G,+2 β,)@J,3[2+
Del valor absoluto:
86. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
! = | + 3| β 2
0 β 3, 50 β | + 3| = + 3
! = + 3 β 2 = + 1
Y: ! = β 10 + 26
= β 10 + 25 + 26 β 25
! = β 5 + 1 β E2)Γ‘G,+2 K*> => 2G)>
Hacia arriba
β - ! =0 β 5, 70
La grΓ‘fica de f(x) es:
! = u
β β 9 ; β5 < β€ β3
+ 1 ; β3 < β€ 5
β 5 + 1 ; 5 < β€ 7
123 ! = 0 β 2, 60
88. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
Sean: ! = β 2 β
E2)Γ‘G,+2 K*> => 2G)> β2'@2 2))@G2
β = 0, 4 = β2
! = β | β 2|
00, 2 . β | β 2| = β + 2
! = + β 2 = 2 β 2
.2, 4 . β | β 2| = β 2
! = β + 2 = 2
! = 2 + β β 4
De: y= 4 + Gβ β β β E2)Γ‘G,+2 K*> => 2G)> β2'@2
+2 L>)>'β2 ; β = 4 , 4 = 2
β = 2 ; 4 = 0
El dominio de f(x) es:
- ! = .β3,0. s .0,2 .s .2, 4.s .4,8 .
- ! = .β3, 8 .
La grΓ‘fica de f es:
89. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
123 ! = [-2, 7]
62)
W = r
+ 10 + 21 ; β .β7, β5. s.β2, β1.
β + 1 + 1 ; β 0 β 1 , 3 0
Sea; W = + 10 + 21 = + 10 + 25 + 21 β 25
W = + 5 β 4
---------- parΓ‘bola que se abre hacia arriba: h=-5, k= -4
De:
β7 β€ < β5 Γ³ β 2 β€ < β1
β2 β€ + 5 < 0 Γ³ 3 β€ + 5 < 4
0 β€ + 5 β€ 4 Γ³ 9 β€ + 5 < 16
β4 β€ + 5 β 4 β€ 0 Γ³ 5 β€ + 5 β 4 < 12
90. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
β4 β€ f(x) β€ 0 Γ³ 5 β€ ! < 12
123 W = .β4,0 . s .5, 12 .
W = 1 + β + 1
-1< x β€ 3
0 < + 1 β€ 4
0 < β + 1 β€ 2
1 < β + 1 + 1 β€ 3
123 W = ]1, 3 ]
123 W = = .β4,0 . s 01, 3 0s .5, 12 .
63)
91. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
| β 2| > 3 β β 2 > 3 Γ³ β 2 < β3
β > 5 Γ³ < β1
! =
N
=
7
= 1 +
7
; x β 2
De:
> 5 Γ³ < β1
β 2 > 3 Γ³ β 2 < β3
AdemΓ‘s: x-2 >0 β
`
> 0
β 2 < β3 β x-2 <0
β < 0
< Γ³ > β
0 < < Γ³ β < < 0
0 <
7
<
7
Γ³ β
7
<
7
< 0
1 < 1 +
7
<
7
+ 1 Γ³ 1 β
7
< 1 +
7
< 1
1< 1 +
7
<
`
Γ³ β
%
< 1 +
7
< 1
1 < ! <
`
Γ³ β
%
< ! < 1
123 ! =0 β
%
, 1. s 01,
`
.
! = : + 4 β 1 = : + 4 + 4 β 5
! = : + 2 β 5
= + 2 β 5
+ 2 β = 5
92. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
O
N
β
O
N
= 1 β β@EΓ©)G,+2
De: 0 < x < 1
2 < + 2 < 3
4 < + 2 < 9
β1 < + 2 β 5 < 4
0 < : + 2 β 5 < 2
123 ! = 00, 2 .
! = 2 + |2 β 5|
.2, 5/2. β |2 β 5| = β2 + 5
! = 2 + 5 β 2 = 7 β 2
.
N
, 30 β |2 β 5| = 2 β 5
! = 2 β 5 + 2 = 2 β 3
! = r
7 β 2 ; 2 β€ < 5/2
2 β 3 ;
N
β€ β€ 3
De: 2 β€ <
N
4 β€ 2 < 5
β5 < β2 β€ β4
2 < 7 β 2 β€ 3
93. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
N
β€ β€ 3
5 β€ 2 β€ 6
2 β€ 2 β 3 β€ 3
123 ! = .2, 30
El rango se la funciΓ³n serΓ‘, la suma de los rangos de las funciones
f1, f2 y f3:
123 ! = ]-4/3 ,1[ U ]1, 10/3[U 00, 2 . s 02, 30
123 ! = 0 β
%
,
`
.
La grΓ‘fica de f, es:
94. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
64)
L1:
[23W 45 = 1 = ?
ΕΈ
β°β°β°β°β° = β°β°β°β°β° ; y = mx+b
P(0,0) β 0 = G β = ?
? = 1 β = ; 0 β€ β€ 2.5
L1: y = ; 0 β€ < 2.5
L2:
Entre A y B la recta es paralela al eje x, por tanto:
= 2.5
L2: y = 2.5 ; 2.5 β€ < 4.5
L3:
1-
β°β°β°β° = J + Β‘-
β°β°β°β° = 3.5
95. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
J + 2.5 + J = 3,5 β J = 0.5
y = mx+b
B(4.5;2.5) β 2.5 = 4.5? + G
C(5,3) β 3 = 5? + G
β B
2.5 = 4.5? + G
β3 = β5? + G
Se obtiene β ? = 1 ; b =2
= β 2
L3: y = β 2 ; 4.5 β€ < 5
L4:
C(5,3) β 3 = 5? + G
D(8,0) β 0 = 8? + G
b = 8
m =-1
= 8 β
L4: y = 8 β ; 5 β€ β€ 8
! = β’
; 0 β€ < 2,5
2.5 ; 2.5 β€ < 4.5
β 2 ; 4.5 β€ < 5
8 β ; 5 β€ β€ 8
65)
96. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
a) ! = 2 β 3 %
+ 5
Si:
! β = β! β β β β@?E2)
! β = ! β β β β β E2)
! β = 2 β β 3 β %
+ 5
= 2 β 3 %
+ 5
! β = ! β β β βE2)
b) ! = 5 β 3 + 1
! β = 5 β β 3 β + 1
= β 5 + 3 + 1
= β 5 β 3 β 1
! β β β! β β β β β 3, @?E2)
! β β ! ----------------- no par
97. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
c) β =
β β = = . 0
= β = ββ
β β = ββ β β β β β @?E2)
d) ! = β2 + 2 + β β2 β 2 +
! β = :2 + 2 β + β β :2 β 2 β + β
! β = β2 β 2 + β β2 + 2 +
! β = βΒ’β2 + 2 + β β2 β 2 + Β£
! β = β! β β β β β βΒ€?E2)
98. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
66)
! = β + 8 β 10
= β β 8 + 10
= β β 8 + 16 + 6
6 β β 4
! β = 6β β β 4
= 6 β .β + 4
= 6 β + 4
! β = β.6 +( + 4 0
De:
! = β β 8 β 10
= β + 8 + 10
= β + 8 + 16 + 6
6 β + 4
! β = 6β β + 4
= 6 β .β β 4
= 6 β β 4
! β = β.6 + β 4 0
99. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
! = P
6 β β 4 ; 2 β€ β€ 6
6 β + 4 ; β6 β€ < 2
La funciΓ³n no es par ni impar
67)
- ! = 0,1,2Q
- W = 0,2,4Q
- ! β© - W = 0, 2Q
- ! + W = 0, 2Q
f+g = , ! 6W / β 0, 2Q
a) ! + W 2 = $2, 0 + &Q = 2,
! + W 2 =
G
!. W 2 = , / = ! . W , β - ! β© - W Q
100. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
- ! β© - W = 0, 2Q
!. W 2 = 0.
1
2
!. W 2 = 0
c.-
(! + 3W 2 = , / = ! + 3W , β - ! β©
- W QQ
(! + 3W 2 = 2, 0 + 3 $ &Q
(! + 3W 2 =
68)
a) ! = | | ; W =
| | = B
, β₯ 0
β , < 0
! + W = ! + W s ! + W
- ! β© - W = β₯ 0 β© 1
- ! β© - W = β₯ 0
- ! β© - W = < 0 β© 1
- ! β© - W = < 0
! + W = B
2 ; β₯ 0
0 ; < 0
101. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
W β ! = B
0 ; β₯ 0
2 ; < 0
b) ! = ; W = B β1,2 , $ ,
%
& , 2, β3 , ~4, β2β’Β₯
! + W =?
- ! = 1 ; - W = Bβ1, , 2, 4Β₯
- ! β© - W = Bβ1, , 2, 4Β₯
! + W = , / = ! + W , β - ! β© - W Q
102. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
! + W = 1, β1 + 2 , $ , +
%
& , 2,2 β 3 , ~4, β2 + 4β’Q
! + W = 1, 1 , Β§
1
2
,
5
4
Β¨ , 2, β1 , ~4, β2 + 4β’Q
W β ! =?
- ! = 1 ; - W = Bβ1, , 2, 4Β₯
- ! β© - W = Bβ1, , 2, 4Β₯
W β ! = , / = W β ! , β - ! β© - W Q
! + W = 1,2 + 1 , $ , %
β & , 2, β3 β 2 , ~4,4 β β2β’Q
! + W = 1, 3 , Β§
1
2
,
1
4
Β¨ , 2, β5 , ~4,4 β β2β’Q
69)
De:
0 β€ β€ 3 β 0 β€ 3 β€ 9
3 < β€ 6 β 9 β€ 3 β€ 18
Se tiene entonces:
! 3 = P
2 , 0 β€ 3 β€ 9 β !
3 , 9 < 3 β€ 18 β !
De:
0 β€ β€ 3 β β2 β€ β 2 β€ 1
103. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
3 < β€ 6 β 1 < β 2 β€ 4
! 3 = P
2 , β 2 β€ β 2 β€ 1 β !
3 , β 1 < β 2 β€ 4 β !%
Sea: W = ! 3 + ! β 2
W = ! + ! s ! + !% s ! + ! s + ! + !%
! + ! = 2 + 2 = 4 ; - ! β© - ! = .0,10
! + !% = 2 + 3 = 5 ; - ! β© - !% = .1.40
! + ! = β
; - ! β© - ! = β
! + !% = β
; - ! β© - !% = β
W = P
4 , β .0,10
5 , β01,40
- W = .0,10s 01, 40
70)
Βͺ
+ ! =? ; g(x) β 0
! = Β’~0, β2β’, ~1, β5β’, 2,0 Β£
W = ~0, β8β’, $2, & , ~4, β3β’Q
104. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
- ! = 0,1,2Q ; - W = 0,2,4Q
- ! β© - W = 0,2Q
Βͺ
= $ ,
Βͺ
& / β - ! β© - W Q
Βͺ
= r$0,
β_
β
& , Ζ2,
β
OΜ
βΒ« ; g(x) β 0
2 β - $
Βͺ
&
Βͺ
= 0, 2 Q
Βͺ
+ ! = $ ,
Βͺ
+ ! & / β - ! β© - W Q
f(2) =0
Βͺ
+ ! = B$0,
β_
β
+ β2 &8
Βͺ
+ ! = 0,4 Q
71)
! + W = ?
Sea:
! = 3 + 4 ; β .0,20
! = 1 β ; β 02, 50
W = ; β .0,3 .
W = 4 ; β .3, 60
105. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
! + W = ! + W s ! + W s ! + W s ! + W
- ! β© - W = .0,20
- ! β© - W = β
β β ! + W
- ! β© - W =02,3.
- ! β© - W = .3,50
! + W = 3 + 4 + = + 3 + 4
! + W = 1 β + = β + 1
! + W = 1 β + 4 = 5 β
! + W = u
+ 3 + 4 , β .0,20
β + 1 , β 02,30
5 β , β .3, 50
- ! = .0,20 s02,3.s.3,50
- ! = .0, 50
La grΓ‘fica es:
106. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
72)
! = β9 β
De: 9 β β₯ 0 β β€ 9
β β3 β€ β€ 3
- ! = .β3, 30
W = 2 β | β 1| ; β 0 β 2, 50
| β 1| = P
β 1 ; β₯ 1
β β 1 ; < 1
a) 0 β 2, β1. β | β 1| = 1 β
W = 2 + β 1 = + 1
b.- .1,50 β | β 1| = β 1
W = 2 β + 1
W = 3 β
W = P
+ 1 , β0 β 2,1.
3 β , β .1,50
! + W = ! + W s ! + W
- ! β© - W = 0 β 2,1.
- ! β© - W = .1,3 0
107. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
! + W = β9 β + + 1
! + W = β9 β β + 3
! + W = r
β9 β + + 1 , β 0 β 2,1.
β9 β β + 3 , β .1,30
73)
! = | β 2| β 1 ; β .β2,6.
W = P
β2 , β .β3,2 .
2 , β .2,6.
)>>=')@G@>3L, ! :
| β 2| = P
β 2 ; β₯ 2
β β 2 ; < 2
108. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
a) 0 β 2,2 . β | β 2| = β β 2
! = β + 2 β 1
! = 1 β
b) .2, 6 . β | β 2| = β 2
! = β 2 β 1
! = β 3
! = P
1 β ; β 0 β 2, 2 .
β 3 , β .2, 6 .
W = P
β2 , β .β3,2 .
2 , β .2,6 .
! + W = ?
! + W = ! + W s ! + W s ! + W s ! + W
Realizar la intersecciΓ³n de dominios:
- ! β© - W = 0 β 2,2. β© .β3, β3. = .β2.2 .
- ! β© - W = 0 β 2,2. β© .2,6 . = β
β β ! + W
- ! β© - W = .2,6. β© .β3,2 . = β
β β ! + W
- ! β© - W = .2, 6. β© .β2,6. = .2, 6 .
! + W = ! + W s ! + W
! + W = 1 β β 2 = β β 1
! + W = β 3 + 2 = β 1
109. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
! + W = P
β β 1 , β .β2, 2 .
+ 1 , β .2, 6 .
+2= W)Γ‘!@'2= L> !, W ! + W =,3:
!:
g:
f+g :
110. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
74)
Sean:
! = + 3 ; ! = 3 + 2
W = 2 β 4 ; W = 2 β
! + W = ! + W s ! + W s ! + W s ! + W
Si las intersecciones de los dominios de las funciones indicadas
existen, las sumas de las funciones existen, caso contrario no
existen.
- ! β© - W = 0 β 4,00 β© .β3, 2 0 = .β3,00
- ! β© - W = 0 β 4,00 β© 02, 8 0 = β
β β ! + W
- ! β© - W = 00, 50 β© .β3, 2 0 =00, 20
- ! β© - W = 00, 50 β©02, 80 =02 , 5.
! + W = + 3 + 2 β 4 = 3 β 1
! + W = 3 + 2 + 2 β 4 = 5 β 2
! + W = 3 + 2 + 2 β = 2 + 4
! + W = u
3 β 1 , β .β3, 00
5 β 2 , β 00, 20
2 + 4 , β 02, 5 .
111. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
75)
W = β 2 ; β₯ β2
>+ L,?@3@, >=:
D(f) = [-2, β .
β = β β 9 βΆ - β =?
De: β 9 β₯ 0
+ 3 β 3 β₯ 0
β β€ β3 Γ³ β₯ 3
- β = ]-β , β30 s .3, β .
! = β | β 1|
Punto crΓtico = {1}
a) X <1 β | β 1| = β β 1
! = + β 1 = 2 β 1
b) xβ₯ 1 β | β 1| = β 1
! = β + 1 = 1
112. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
! = B
1 , β₯ 1
2 β 1 , < 1
- ! + W = - ! β© - W
! + W = ! + W s ! + W
- ! β© - W = .1, β . β© .β2, β . = .1, β.
- ! β© - W =0 β β, 1 . β© .β2, β .β2,1 .
! + W = P β 2 + 1
β 2 + 2 β 1
! + W = P
β 1 , β₯ 1
+ 2 β 3 , β2 β€ < 1
- ! + W = .β2, 1 . s .1, β .
= .β2, β .
- ! + W . β0 = - ! + W β© - β
Como: - β = ]-β , β30 s .3, β .
- ! + W . β0 = .β2, β . β© ]-β , β30 s .3, β .Q
- ! + W . β0 = [3, β .
76)
! = ; W = |2 |
113. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
Como: |2 | = B
2 ; β₯ 0
β2 ; < 0
! + W = ! + W s ! + W
Analizar las intersecciones de los dominios:
- ! β© - W = 1 β© .0, β. = .0, β .
- ! β© - W = 1 β© 0 β β, ,. =0 β β, 0 .
! + W = + 2 = + 2 + 1 β 1 = + 1 β 1
! + W = β 2 = β 2 + 1 + 1 = β 1 + 1
! + W = P
+ 2 ; β .0, β .
β 2 ; β 0 β β, 0 .
La grΓ‘fica de las parΓ‘bolas son:
- ! + W = 1
123 ! + W = .0, β .
77)
114. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
!: 0 β 3,50 + .β5,30 / ! = β + 2 + 3
β + 2 + 3 = β β 2 + 1 + 4
= 4 β β 1
W = β9 β
Si: 9 β β₯ 0
β€ 9 β β3 β€ β€ 3
- W = .β3, 30
- ! β© - W = 0 β 3,50 s .β5, 30 β© .β3,30 = .β3,30
β β
Βͺ
Βͺ
=
% O
β6 O
β9 β > 0
9 β > 0
< 9 β β3 < < 3
- $
Βͺ
& = 0 β 3,3.
115. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
78)
! = β 2 + 2 ; β₯ 5
W = 2| β 1| + 1 ; β .β3,4.
| β 1| = P
β 1 , β₯ 1
β β 1 , < 1
.β3, 1 . β | β 1| = β β 1
W = β2 β 1 + 1
W = 3 β 2
.1, 4. . β | β 1| = β 1
W = 2 β 1 + 1
W = 2 β 1
W = P
3 β 2 , β .β3,1.
2 β 1 , β .1, 4.
! + W = ! + W s ! + W
Se debe determinar el dominio de f:
! = β 2 + 2 ; β₯ 5
5= ( β 2 + 1 + 1
5 = β 1 + 1
116. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
4 = β 1
β 1 = β 2
= 3 ; = β1
- ! = 0 β β, β10 s .3, β .
! + W β - ! β© - W = 0 β β, β10s .3, β .Q β© .β3,1.
- ! β© - W = .β3, β10
! + W = β 2 + 2 + 3 β 2
! + W = β 4 + 5 = β 4 + 4 + 1
! + W = β 2 + 1
! + W β - ! β© - W = 0 β β, β1s.3, β . Q β© .1, 4.
- ! β© - W = .3, 4.
! + W = β 2 + 2 + 2 β 1
! + W = + 1
! + W = P
β 2 + 1, β .3, β10
+ 1 , β .3,4.
Su rango es:
117. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
-3 β€ x β€ -1
β5 β€ β 2 β€ β3
9 β€ β 2 β€ 25
10 β€ β 2 + 1 β€ 26
123 W = 010,260
3 β€ < 4
9 β€ < 16
10 β€ + 1 < 17
123 W = .10, 17 .
123 ! + W = .10,260
79)
Β― + 4Β° = 4 ; β₯ 0
= 1 β
%
β
%
< 0 β β β βE2)Γ‘G,+2 => 2G)> 2 +2 @JK*@>)L2
β = 1 ; 4 = 0
- ! = 0 β β, 10
118. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
123 ! = .0, β .
80)
4 β = 144
O
^
β
O
%%
= 1 β β β β@EΓ©)G,+2
2 = 6 ; G = 12
AsΓntotas:
4 β = 0
2 β 2 + = 0
2=Γ3[,[2=: P
= 2
= β2
- ! = 1
)23 ! = 0 β β, 60 s .6, β .
119. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
81)
! = + 2 ; β .β1, 2.
β1 β€ < 2
0 β€ < 4
2 β€ + 2 < 6
123 ! = .2, 6 .
82)
! = + 4 β 1
! = + 4 + 4 β 5
! = + 2 β 5 ------parΓ‘bola que se
Abre hacia arriba.
β = β2 ; 4 = β5 ; Y β2, β5
De:
β2 < β€ 3
120. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
0 < + 2 β€ 5
0 < + 2 β€ 25
β5 < + 2 β 5 β€ 20
123 ! = 0 β 5, 20 0
83)
! = 3 + 2 β ; β .β2,2.
! = β β 2 + 1 + 4
! = 4 β β 1
β2 β€ < 2
β3 β€ β 1 < 1
0 β€ β 1 β€ 9
β9 β€ β β 1 β€ 0
β5 β€ 4 β β 1 β€ 4
Ran(f) = [-5, 4[
121. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
84)
! = β 2 ; ! = + 5
! = 2 β 4
W = ; W = 3
β = ! W + ! W + ! W + ! W + ! W + ! W
- ! β© - W = .β4,20 β© 00,2. = 00,20
- ! β© - W = .β4,20 β© .2,8. = β
- ! β© - W =02,60 β© 00,20 = β
- ! β© - W =02,60 β© .2,8. = .2,6 0
- ! β© - W =06, 90 β© 0, 20 = β
- ! β© - W =06, 90 β© .2,8 . =06,8.
β = ! W + ! W + ! W
! W = β 2 . = β 2 %
! W = ( + 5 . 3 = + 15
! W = 3 2 β 4 = 6 β 12
122. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
β = β’
β 2 %
, β 00,2 0
3
2
+ 15 , β02,60
6 β 12 , β06,8.
b)
Βͺβ
β
=
Ε
O
ΒͺO
O
= Β±
O
N
=
^
`
ΒͺO
Ε
=
%
β =
β©
βͺ
β¨
βͺ
β§
β 2
, β 00,2 0
6
+ 10
, β02,60
3
2 β 4
, β06,8.
85)
! = P
| β 2|| + 2| , β .β6,00
2 , β₯ 2
123. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
a) .β6, β2. β | β 2| = β β 2
| + 2| = β + 2
! = β 2 + 2 = β 4
a) [-2,0] β | β 2| = β β 2
| + 2| = + 2
! = β β 2 + 2 = 4 β
! = u
β 4 , β .β6, β2.
4 β , β .β2,00
2 , β₯ 2
W = B
+ 2 , β₯ β2
1 , < β2
Βͺ
= ?
Βͺ
=
β
Βͺβ
+
β
ΒͺO
+
O
Βͺβ
+
O
ΒͺO
+
Ε
Βͺβ
+
Ε
ΒͺO
Determinar las intersecciones de los dominios para la
existencia de las funciones:
- ! β© - W = .β6, β2. β© .β2, β. = β
- ! β© - W = .β6, β2. β©0 β β, β2. =0 β 6, β2.
- ! β© - W = .β2,00 β© .β2, β. = .β2,00
- ! β© - W = .β2, 00. β© 0 β β, β2. = β
- ! β© - W = .2, β . β© .β2, β. = .2, β.
124. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
- ! β© - W = .2 , β . β© .β β, β3. = β
Se tiene que:
Βͺ
=
β
ΒͺO
+
O
Βͺβ
+
Ε
Βͺβ
β
ΒͺO
=
O %
= β 4
O
Βͺβ
=
% O
= 2 β
Ε
Βͺβ
=
Βͺ
= β’
β 4 , β .β6, β2.
2 β , β .β2,00
, β .2, β .
123 ! = .0, β .
125. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
86)
! = ; β 2 ; W = ; β 0
!,W = !.W 0 = ! $ &
!.W 0 = Β±βΕ
Β±
= ; β β1
- !,W = 1 β β1, β2,0Q
W,! = W.! 0 = W $ &
W.! 0 =
β
Β±βO
β
Β±βO
=
ΕΒ±βΒΆ
Β±βO
β
Β±βO
=
7
= 3 + 7
- W,! = 1-{-2,0}}
- !,W β© - W,! = R-{-1,-2,0}
87)
126. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
gof =g[f(x)]
W = = 2 +
! = ; β₯ 3
- ! = .3, β.
SI: β₯ 3
β 2 β₯ 1 β β 2 > 0
β >0
β 2 β₯ 1
β€ 1 β 0 < β€ 1
123 ! β© - W =00, 1. β© . , β.= , 1. β β
β β W,!
123 ! β - W = ?
123 ! β - W β - W,! = / β - ! ! β - W Q
127. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
- W,! = β₯ 3 β§ β₯
β β₯ 3 β§ β β₯ 0
β β₯ 3 β§ β₯ 0
β β₯ 3 β§
%
β₯ 0
β β₯ 3 β§ 4 β β 2 β₯ 0
β β β₯ 3 β§ 2 β€ β€ 4
β .3,40
- W,! = β .3, 4 0
88)
! = 2 β 3
W = + 1
- W = 1 ; 123 W = .1, β .
- ! = 1 ; 123 ! = 1
128. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
!,W = !.W 0 = ! + 1
!.W 0 = 2 + 1 β 3 = 2 β 1
123 W β© - ! = .1, β. β© 1 = .1, β .
β β !,W
W,! = W.! 0 = ! 2 β 3
W.! 0 = 2 β 3 + 1
W.! 0 = 4 β 12 + 10
123 ! β© - W = 1 β© 1 = 1
β β W,!
De: W,! = !,W
2 β 1 = 4 β 12 + 10
2 β 12 + 11 = 0
=
Β±β %% __
%
=
Β±βN^
%
La suma de los valores de x:
S= 3 +
βN^
%
+ 3 β
βN^
%
w = 6
89)
129. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
!,W = + + 1
W = + 1
!.W 0 = + + 1 β ! + 1 = + + 1
[ = + 1 ; = β[ β 1
Ε
! [ = β[ β 1
Ε
+ [
! = β β 1
Ε
+
W,! = W.! . = ~β β 1
Ε
+ β’ + 1
= β 1 + 3 β β 1
Ε
+ 3 ~β β 1
Ε
β’ + 1 +
= + 3 β β 1
Ε
+ 3 ~β β 1
Ε
β’ +
W,! 9 = 9 + 243β9 β 1
Ε
+ 27 β9 β 1
Ε
+ 9
= 9 +729+243(2)+108
W,! 9 = 1332
90)
! β 1 = 3 + 2 + 12
W + 1 = 5 + 7
Se halla las funciones f(x) y g(x):
130. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
[ = β 1 ; = [ + 1
! [ = 3 [ + 1 + 2 [ + 1 + 12
! [ = 3[ + 6[ + 3 + 2[ + 2 + 12
! [ = 3[ + [ 6 + 2 + 2 + 15
! = 3 + 6 + 2 + 2 + 15
Sea: [ = + 1 ; = [ β 1
W [ = 5 [ β 1 + 7
W [ = 5[ + 2
W = 5 + 2
La funciΓ³n compuesta fog es:
!,W = !.W 0 = ! 5 + 2
!.W 0 = 3 5 + 2 + 5 + 2 6 + 2 + 2 + 15
= 75 + 60 + 12 + 52 + 30 + 22 + 12 + 2 + 15
= 75 + 90 + 52 + 39 + 32
Si: !,W β2 = β42
!,W β2 = 75 4 + 90 β2 β 102 + 39 + 32 = β42
159 β 72 = β42
159 = 32
2 = 53
131. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
91)
r
! = β2 β 1
W = β2 β 7
De: !,β = W
!,β = !~β β’
!.β 0 = β2 β 7
β = [
! [ = β2[ β 1 = β2 β 7
2[ β 1 = 2 β 7
[ = β 3
β β = β 3
92)
De: ! β 2 =
[ = β 2 ; = [ + 2
! [ =
ΒΉ
=
ΒΉ
132. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
! =
De: !,! $ & = 5, => [@>3>:
!,! = !.! 0 = ! $ &
!.! 0 = O
Β±ΒΊβ
= OΒΊΒ±ββ
Β±ΒΊβ
=
!,! $ & =
$
O
Β±
&
O
Β±
= 5
%
= 5 β 4 β 2 = 15 β 10
17 = 14
=
%
7
93)
!,W = 2 +16x+25
! W 0 = 2 +16x+25
Sea: g(x) = u
! * = 2 +16x+25 ------(a)
De:
! = 2 β 4 β 5 β ! * = 2* β 4* β 5 ---(b)
De (a) y (b):
133. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
2 +16x+25 = 2* β 4* β 5
2* β 4* β 2 + 16 +30) =0
* =
%Β± : ^ _ O ^ `
%
=
%Β±β ^ O _ N^
%
* = 1 Β±
%
%
β + 8 + 16
* = 1 Β± : + 4
* = 1 + | + 4|
u= g(x)
W = P
+ 5 , β₯ β4
β β 3 , < β4
94) Si, f(x)= + 2 + 2 , β2++2) W , =@:
!,W = β 4 + 5
De:
!.W 0 = β 4 + 5
!.W 0 = .W 0 + 2W + 2
.W 0 + 2W + 2 = β 4 + 5
.W 0 + 2W β β 4 + 3 = 0
W =
Β±:% % O %
W = β1 Β± β β 4 + 4
W = β1 Β± : β 2
W = β1 + | β 2|
W = B
β 3 , β₯ 2
β + 1 . < 2
134. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
94)
Sean: ! = ; ! = β
W = β ; W = 2
W, ! = (W ,! + W ,! + W ,! + W ,!
El rango de la funciΓ³n βfβes:
sI; < 1
β₯ 0 β β β β123 ! = .0, β.
Si: β₯ 2 β
β β€ 8 β β β€ β8
123 ! =0 β β, β8 0
W ,! :
123 ! β© - W = .0, β. β© 0 β β, 2.= .0,2.
123 ! β - W = ?
123 ! β - W β - W ,! = / β - ! β§ ! β
- W
- W ,! = < 1 β§ < 2
135. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
< 1 β§ | | < 2
< 1 β§ β β2 < < β2
β 0 β β2 , 1 .
- W ,! = 0 β β2 , 1 .
W ,! :
123 ! β© - W =0 β β, β80 β©0 β β, 2.= ββ, β80
123 ! β - W = ?
123 ! β - W β - W ,! = - ! = .2, β .
W ,! :
123 ! β© - W = .0, β β© .4, β.= .4, β.
123 ! β - W = ?
123 ! β - W β - W ,! = / β - ! β§ ! β
- W
- W ,! = < 1 β§ β₯ 4
< 1 β§ β 2 + 2 β₯ 0
< 1 β§ β€ β2 Γ³ β₯ 2 Q
β 0 β β , β2 0
W ,! :
123 ! β© - W = .β β, β8 β© .4, β.= β
β β W ,!
Finalmente:
- W,! = 0 β β2 , 1 . s .2, β .s 0 β β , β2[
- W,! = 0 β β , β2[ U0 β β2 , 1 . s .2, β .
136. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
95)
W,! = W.! 0
Sean: ! = + 1 ; ! = β
W = β 1 ; W = 2
W, ! = (W ,! + W ,! + W ,! + W ,!
El rango de la funciΓ³n βfβes:
< 1 β β₯ 0
+ 1 β₯ 1
! β₯ 1
123 ! = .1, β .
β₯ 4 β β₯ 16
β β€ β16
123 ! =0 β β, β160
W ,! :
123 ! β© - W = .1, β. β© 0 β β, 2.= .1,2.
137. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
123 ! β - W = ?
123 ! β - W β - W ,! = / β - ! β§ ! β
- W
- W ,! = < 1 β§ + 1 < 2
< 1 β§ | | < 1
< 1 β§ β 1 < < 1
β 0 β 1 , 1 .
- W ,! = 0 β 1 , 1 .
W ,! :
123 ! β© - W =0 β β, β160 β©0 β β, 2.= ββ, β160
123 ! β - W = ?
123 ! β - W β - W ,! = - ! = .4, β .
W ,! :
123 ! β© - W = .1, β β© .4, β.= .4, β.
123 ! β - W = ?
123 ! β - W β - W ,! = / β - ! β§ ! β
- W
- W ,! = < 1 β§ + 1 β₯ 4
< 1 β§ β β3 + β3 β₯ 0
< 1 β§ Β’ β€ ββ3 Γ³ β₯ β3 Β£
β 0 β β , ββ3 0
W ,! :
123 ! β© - W = .β β, β16 β© .4, β.= β
138. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
β β W ,!
Finalmente:
- W,! = 0 β 1 , 1 . s .4, β .s 0 β β , ββ3[
- W,! = 0 β β , ββ3[ U0 β 1 , 1 . s .4, β .
96)
!,W = !.W 0
Sea: W = 1 β ; W = 2
!,W = !,W + !,W
Se debe determinar el Rango de g:
Si: X <-2
β > 2
1-x > 3 β W > 3
123 W = .3, β.
Si: > 6
2 > 12 β W > 2
139. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
123 W =012, β.
!,W :
123 W β© - ! = ?
123 W β© - ! = .3, β. β© 0 β 2,20.=03,20.
123 W β - ! = ?
123 W β - ! β - !,W = / β - W β§ W β
- !
- !,W = < β2 β§ β2 < 1 β < 20
< β2 β§ β3 < β < 19
< β2 β§ β19 < < 3
β 0 β 19, β2.
- !,W = 0 β 19, β2.
(!,W = !.W 0 = ! 1 β
!.W 0 = 2 1 β + 1
= 2 β 4 + 2 + 1
!.W 0 = 2 β 4 + 3
!,W :
123 W β© - ! = ?
123 W β© - ! = .12, β. β© 0 β 2,20.=012,20.
123 W β - ! = ?
123 W β - ! β - !,W = / β - W β§ W β
- !
- !,W = > 6 β§ β2 < 2 < 20
140. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
> 6 β§ β1 < < 10
> 6 β§ β1 < < 10
β 06, 10.
- !,W = 06,10.
(!,W = !.W 0 = ! 2
!.W 0 = 2 2 + 1
= 8 + 1
!.W 0 = 8 + 1
!,W = P
2 β 4 + 3 , β0 β 19, β2.
8 + 1 , β 06,10 .
97)
!,W = !.W 0
Sea: W = 2 ; W = β3
! = 3 + 2
141. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
!,W = !,W + !,W
Se debe determinar el Rango de g:
Si: x < 0
2 < 0
β W < 0
123 W =0 β β, 0.
Si: β₯ 1
3 β₯ 3 β β3 β€ β3
β W β€ β3
123 W =0 β β, β30
!,W :
123 W β© - ! = ?
123 W β© - ! =0 β β, 0. β© 0 β β, β3.=0 β β, β3.
123 W β - ! = ?
123 W β - ! β - !,W = / β - W β§ W β
- !
- !,W = < 0 β§ 2 < β3
< 0 β§ < β
β 0 β β, β .
- !,W = 0 β β, β .
(!,W = !.W 0 = ! 2
!.W 0 = 3 2 + 2
= 6 + 2
142. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
!.W 0 = 6 + 2
!,W :
123 W β© - ! = ?
123 W β© - ! = .ββ, β3. β© 0 β β, β3.=0 β β, β3.
123 W β - ! = ?
123 W β - ! β - !,W = - W
- !,W = .1, β .
- !,W = .1, β .
(!,W = !.W 0 = ! β3
!.W 0 = 3 β3 + 2
= 2 β 9
!.W 0 = 2 β 9
!,W = r
6 + 2 , β 0 β β, β .
2 β 9 , β .1, β .
98)
143. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
!,W = !.W 0
Sean: ! = + 2 ; ! = β 1
W = ; W = 1 β
!, W = (! ,W + ! ,W + ! ,W + ! ,W
El rango de la funciΓ³n βgβes:
< 0 β β₯ 0
W β₯ 0
123 W = .0, β .
β₯ 0 β β₯ 0
β β€ 0
1 β β€ 1
123 W =0 β β, 10
! ,W :
123 W β© - ! = .0, β. β© 0 β β, 10 = .0,1.
123 W β - ! = ?
123 W β - ! β - ! ,W = / β - W β§ W β
- !
- ! ,W = < 0 β§ β€ 1
< 0 β§ | | β€ 1
< 0 β§ β 1 β€ β€ 1
β .β1 ,0 .
- ! ,W = .β1 , 0 .
! .W 0 = !
= + 2
144. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
! .W 0 = + 2
! ,W :
123 W β© - ! =0 β β, 10 β©0 β β, 1.= ββ, 10
123 W β - ! = ?
123 W β - ! β - ! ,W = - W = .0, β .
! .W 0 = ! 1 β
= 1 β + 2
! .W 0 = 3 β
! ,W :
123 W β© - ! =00, β . β© .1, β.= .1, β.
123 W β - ! = ?
123 W β - ! β - ! ,W = / β - W β§ W β
- !
- ! ,W = < 0 β§ > 1
< 0 β§ β 1 + 1 > 0
< 0 β§ > β1 Γ³ > 1 Q
β 0 β β, β1.
! .W 0 = !
= β 1
! .W 0 = β 1
! ,W :
123 W β© - ! = .β β, 10 β©01, β.= β
145. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
β β ! ,W
Finalmente:
- !,W = .β1 , 0 . s .0, β .s 0 β β , β1[
- W,! = 0 β β , β1[ U.β1 , 0 . s .0, β .
!,W = u
β 1 , β 0 β β , β1 .
+ 2 , β .β1,0.
3 β , β .0, β.
99)
!,W = !.W 0
Sean: ! = β 3 ; ! = 3 β
W = 3 β ; W = 5 β
!, W = (! ,W + ! ,W + ! ,W + ! ,W
El rango de la funciΓ³n βgβes:
β€ 1 β β β₯ β1
β 3 β β₯ 2
W β₯2
146. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
123 W = .2, β .
Si: > 1 β β < β1
5 β < 4
W < 4
123 W =0 β β, 4 .
! ,W :
123 W β© - ! = .2, β. β© 0 β β, 30 = .2,30
123 W β - ! = ?
123 W β - ! β - ! ,W = / β - W β§ W β
- !
- ! ,W = β€ 1 β§ 3 β β€ 3
β€ 1 β§ β β€ 0
β€ 1 β§ β₯ 0
β .0, 10
- ! ,W = . 0, 10
! .W 0 = ! 3 β
= 3 β β 3 3 β
! .W 0 = 9 β 6 + β 9 + 3
! .W 0 = β 3
! ,W :
123 W β© - ! = .2, β . β©0 β β, 3.= .2,3.
123 W β - ! = ?
123 W β - ! β - ! ,W = / β - W β§ W β
- !
147. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
X > 1 β§ 5 β β€ 3
> 1 β§ β β€ β2
> 1 β§ β₯ 2
β .2, β .
- ! ,W = . 2, β .
! .W 0 = ! 5 β
= 5 β β 3 5 β
= 25-10 + β 15 + 3
! .W 0 = β 7 + 10
! ,W :
123 W β© - ! =02, β . β© .3, β.= .3, β.
123 W β - ! = ?
123 W β - ! β - ! ,W = / β - W β§ W β
- !
- ! ,W = β€ 1 β§ 3 β > 3
β€ 1 β§ β > 0
β€ 1 β§ < 0
β 0 β β, 0.
! .W 0 = ! 3 β
= 3 β 3 β
= 3 β 9 + 6 β
! .W 0 = β + 6 β 6
148. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
! ,W :
123 W β© - ! = .β β, 4. β©03, β.=03,4.
123 W β - ! = ?
123 W β - ! β - ! ,W = / β - W β§ W β
- !
- ! ,W = > 1 β§ 5 β > 3
> 1 β§ β > β2
> 1 β§ < 2
β 01, 2.
! .W 0 = ! 5 β
= 3 β 5 β
= 3 β 25 + 10 β
! .W 0 = β + 10 β 22
Finalmente:
!,W =
β©
βͺ
β¨
βͺ
β§ β 3 , β .0,1 0
β 7 + 10 , β .2, β.
β + 6 β 6 , β0 β β, 0.
β + 10 β 22 , β 01,2.
100)
149. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
!,W = !.W 0
Sean: ! = β1 β ; ! =
W = β 4 ; W = 0
!, W = (! ,W + ! ,W + ! ,W + ! ,W
El rango de la funciΓ³n βgβes:
0 β€ β€ 4 β 0 β€ β€ 16
β β4 β€ β 4 β€ 12
β4 β€ W β€ 12
123 W = .β4,120
123 W = 0
! ,W :
123 W β© - ! = .β4,120 β©0 β 3,1.= .β3, β1.
123 W β - ! = ?
123 W β - ! β - ! ,W = / β - W β§ W β
- !
- ! ,W = 0 β€ β€ 4 β§ β3 < β 4 < 1
0 β€ β€ 4 β§ 1 < < 5
0 β€ β€ 4 β§ 1 < < β5
150. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
β01, β5.
- ! ,W =0 1, β5 .
! .W 0 = ! β 4
= :1 β β 4
! .W 0 = β5 β
! ,W :
123 W β© - ! = 0 β©0 β 3,1. = 0Q
123 W β - ! = ?
123 W β - ! β - ! ,W = - W
- ! ,W =04,7.
! .W 0 = ! 0
= β1 β 0
= 1
! .W 0 = 1
! ,W :
123 W β© - ! = .β4,120 β© .3,80 = .3,80
123 W β - ! = ?
123 W β - ! β - ! ,W = / β - W β§ W β
- !
- ! ,W = 0 β€ β€ 4 β§ 3 β€ β 4 β€ 8
0 β€ β€ 4 β§ 7 β€ β€ 12
0 β€ β€ 4 β§ β7 β€ β€ β12
β .β7, :120
151. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
- ! ,W = .β7 , 2β30
! .W 0 = ! β 4
! .W 0 = O %
! ,W :
123 W β© - ! = 0 β© .3,80 = β
β β ! ,W
Finalmente:
!,W = β’
β5 β , β01, β5 .
1 , β 04,7 .
O %
, β .β7, 2β3 0
101)
!β
= !*3'@Γ³3 @3t>)=2
Determinar las inversas de f y de g:
! =
^
%
; β 4
β 4 = 2 + 6 β β 4 = 2 + 6
β 2 = 4 + 6
β 2 = 4 + 6 β =
% ^
Intercambiando las variables x e βyβ:
152. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
!β
=
% ^
; β 2
W = ; β 0
2 = + 2 β 2 = + 2
2 β = 2
β =
Intercambiando las variables x e βyβ:
Wβ
= ; β
!β
,W = !β.W 0
!β.W 0 = !β
$ &
=
4 $
2
2 β 1& + 6
2
2 β 1
β 2
!β.W 0 =
% %
=
^
De: (!β
,W 2 = 6
^]
]
= 6
18 a =11 β 2 =
_
Si: 3 = Wβ
,! 2 +
^
7
2 +
^
7
=
_
+
^
7
=
^N
N%
Wβ
,!
^N
N%
= ?
Wβ
,! = Wβ.! 0
153. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
Wβ.! 0 = Wβ
$
^
%
&
=
$
OΒ±βΛ
Β±ΒΊβ
&
= βΒ±ββO
Β±ΒΊβ
=
%
% %
Wβ.! 0 =
_
^
Wβ
,! $
^N
N%
& =
β
Λβ’
β’β
_
β
Λβ’
β’β
^
=
ΒΊΕβO
β’β
βββ’ΒΌ
β’β
= β
`
`N6
102)
! = ; β 2
W = ; β 2
Wβ
,! * = 3
De; W =
β 2 = + 3
β = 2 + 3
β 1 = 2 + 3
=
La inversa Wβ
>=:
Wβ
= ; β 1
De; ! =
β 2 = 3
154. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
β 3 = 2
β 3 = 2
=
La inversa !β
>=:
!β
= ; β 3
(Wβ
,! = Wβ
$ &
(Wβ
,! =
$
ΕΒ±
Β±ΒΊO
&
ΕΒ±
Β±ΒΊO
=
ΛΒ±βΕΒ±ΒΊΛ
Β±ΒΊO
ΕΒ±ΒΊΒ±βO
Β±ΒΊO
(Wβ
,! =
6 ^
Entonces: (Wβ
,! * =
6Β½ ^
Β½
De: (Wβ
,! * = 3
6Β½ ^
Β½
= 3 β 9* β 6 = 6* + 6
3* = 12 ; * = 4
Se calcula: !β
,W * + 2 = ?
* + 2 = 6
!β
,W 6 = !β.W 0 6
!β.W 0 = !β
$ & =
Β±βΕ
Β±ΒΊO
Β±βΕ
Β±ΒΊO
=
^
6
155. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
!β.W 0 =
^
6
!β.W 0 6 =
^ ^
6 ^
=
_
= β6
!β
,W * + 2 = β6
103)
! = 3 + 5
W = 2 + G
! W 0 = β β 1
De: y = 3x+5
=
N
La inversa de f, es:
!β
=
N
Si: W 0 = β ! 2 + G =
3(ax+b)+5 =x
32 + 5 + 3G = β B
32 = 1
5 + 3G = 0
2 = ; G = β
N
W = β
N
Se tiene que: !β
,W = !β.W 0 = !β
$ β
N
&
!β.W 0 =
β
Ε
β’
Ε
N
=
Β±ΒΊβ’ΒΊββ’
Ε
156. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
!β.W 0 =
6
β 20
]
+ 5 = + 5 =
^
(!β
,W $
^
& =
6
$
^
β 20& =
^ ^`
7
!β
,W $
^
& = β
%%
7
104)
! =
%]
N
!β
3 = 22 β 36
!β
5 = 32 + G
La inversa de f, es:
=
%]
N
β 5 = 3 β 42
=
N %]
!β
=
N %]
!β
3 =
N %]
= 22 β 36
15 + 42 = 62 β 108
22 = 123 ; 2 =
!β
5 =
N N %]
= 32 + G
157. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
25+4(a) =9 a+3b
25 β 52 = 3G
3G = 25 β 5 $ & =
N^N
Se tiene que:
2 β 3G = +
N^N
= 344
3 = !β
2 β 3G = ?
!β
=
N %]
=
N %^
3 = !β
344 =
N %% %^
3 =
6^^
105)
D(f) = [1, 4]
, β - ! , ! = ! β =
β 2 + 3 = β 2 + 3
β 2 = β 2
β β 2 + 2 = 0
β + β 2 β = 0
β + β 2 = 0
158. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
Si: β = 0 β =
Si: + = 2
, β .1,40: βΆ = 1 βΆ = 1
+ = 2 β =
β ! β β β β β >= @3 >'[@t2
! β β β G@ >'[@t2 β P
@3 >'[@t2
=,G)> >'[@t2
! = β 2 + 3 = β 1 + 2
Ran (f) = [a,b]
= .2, G0
De:
1 β€ β€ 4
0 β€ β 1 β€ 3
0 β€ β 1 β€ 9
2 β€ β 1 β€ 11
123 ! = .2,110
= .2,110
!; .1,40 β .2,110
106)
159. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
a) ! = 2| | β
A1. β₯ 0 β | | =
! = 2 β =
A2. X< 0 β | | = β
! β 2 β = β3
! = B
, β₯ 0
β3 , < 0
! =
, β - ! , ! = ! β =
=
β₯ 0 β ! β₯ 0
123 ! = .0, β .
! = β3
, β - ! , ! = ! β =
β3 = β3
β = β
=
< 0 β β β₯ 0
β3 β₯ 0
123 ! = .0, β .
Como: 123 ~! β’ β© 123 ~! β’ = .0, β. β β
β ! β β β β3, >= @3 >'[@t2
a) ----------- (V)
160. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
b) W =
, β - ! , ! = ! β =
β
β
= O
O
β β 2 + β 2 = + β 2 β 2
β2 + = β 2
β + 2 β 2 = 0
3 = 3
= -------------inyectiva
Sobreyectiva:
= β β 2 = + 1
= ; β 1
De: ! = ! $ &
! =
OΒΏββ
ΒΏΒΊβ
OΒΏββ
ΒΏΒΊβ
= =
! β
----------------no es sobreyectiva
c) β = 2 + 3 >= @3 >'[@t2
, β - ! , ! = ! β =
2 + 3 = 2 + 3
2 = 2
= β β β β@3 >'[@t2
161. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
107)
Sean las funciones inyectivas:
! = 3 β 6 + 4
W =
Se determina las funciones inversas de f y g:
= 3 β 2 + 1 + 4 β 3
= 3 β 1 + 1
3 β 1 = β 1
β 1 = β 1
β 1 = Β±M β = 1 Β± M
De: x> 1
β 1 > 0
3 β 1 > 0
3 β 1 + 1 > 1
123 ! = .1, β.
Y > 1 β = 1 + M
!β
= 1 + M
De: =
162. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
+ = β 2
1 β = + 2
=
Intercambiando las variables:
= ; β 1
Ran (g) = R-{1}
Wβ
=
Se conoce que: !β.Wβ
2 0 = 2 ,
!β.Wβ 0 = !β
!β.Wβ 0 = 1 + M
Β±βO
βΒΊΒ±
= 1+ M
!β.Wβ
2 0 = 1 + M
]
]
= 2
M
]
]
= 1 β
]
]
= 1
22 + 1 = 3 β 32 β 2 =
N
De: 3 = ! 5W $2 +
_
N
&8 = ! 5W $N
+
_
N
&8 = !.W 2 0
3 = ! $ & = ! 0
3 = 3(0-1 + 1
163. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
3 = 4
! 5W $2 +
_
N
&8 = 4
108)
!: β ! = , = .β1,40
! = P
5 β 3 , β .β1,2.
3 β 6 + 12 , β .2,40
a) f es biyectivaβ¦.?
21.
! = 5 β 3
, β .β1,2 .; ! = ! β =
5 β 3 = 5 β 3
- = β
= β β β β β @3 >'[@t2
22.
! = 3 β 6 + 12
, β .2,40; ! = ! β =
3 β 6 + 12 = 3 β 6 + 12
3 β 6 = 3 β 6
β 2 = β 2
164. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
β β 2 + 2 = 0
β + β 2 β = 0
β + β 2 = 0
w@: β .2,40 β + β 2 β 0
β β = 0 β =
= β β β β β @3 >'[@t2
De: β .β1,2. β β1 β€ < 2
β3 β€ 3 < 6
β6 < β3 β€ 3
β1 < 5 β 3 β€ 8
123 ! =0 β 1,80
Reescribiendo a : 3 β 6 + 12
3 β 6 + 12 = 3 β 2 + 1 + 12 β 3
= 3 β 1 + 9
De: β .2,4. β 2 β€ < 4
1 β€ β 1 < 3
1 β€ β 1 < 9
3 β€ 3 β 1 < 27
12 β€ 3 β 1 + 9 < 36
123 ! = .12, 36.
123 ! β© 123 ! = β
165. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
β ! β β β β β β β >= @3 >'[@t2
b) B= ]-1, 36] β¦..?
Como: 123 ! = .! 2 , ! 4 . = .12,36.
]-1, 36] β .12,36.
β 0 β 1,360 β β β β β β!2+=,
c) !β
10 = 1 +
β
β¦ . ?
! β β β @3 >'[@t2 β β !β
! = 5 β 3 β = 5 β 3
β 3 = 5 β
=
N
β =
N
!β
=
N
------
! = 3 β 1 + 9 β = 3 β 1 + 9
3 β 1 = β 9
β 1 =
6
= 1 Β±
6
β€ 36 β = 1 + M
6
!β
= 1 + M
6
β !β
10 =
N `
= β
N
166. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
!β
10 = 1 + M
` 6
= 1 +
β
β !β
10 β 1 +
β
d) !β
4 + !β
21 =
`
β¦ . ?
!β
= β’
N
, β .β1,8.
1 + M
6
, β .12,36.
!β
4 =
N %
=
!β
21 = 1 + M
6
= 1 + 2 = 3
!β
4 + !β
21 = + 3
!β
4 + !β
21 =
`
β β β β Y
109)
! =
| |
; W =
Si; xβ₯ 0 β | | =
! =
< 0 β | | = β
167. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
! =
! = u
, > 0
, < 0
, > 0; ! = ! β =
! =
β
β
= O
O
+ = +
= β β β β β @3 >'[@t2
! β @3 >'[@t2
, < 0; ! = ! β =
! =
β
β
= O
O
β = β
= β β β β β @3 >'[@t2
! β @3 >'[@t2
De: ! = = 1 +
x>0 β > 0
1 + > 1
123 ! = 01, β .
! = = β 1
168. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
X < 0 β < 0
1 + < 1
123 ! = 0 β β, 1 .
123 ! β© 123 ! = β
β ! β β β β β @3 >'[@t2
De: W =
, β - W ; ! = ! β =
β
=
O
= β β β β β @3 >'[@t2
W β β β β β β@3 >'[@t2
Si:
! = ; = 1 +
β 1 = 1
= β =
La inversa de ! , >=:
!β
= , β 1
! = ; = 1 β
+ 1 = 1
= β =
La inversa de ! , >=:
169. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
!β
= , β β1
W = ; y= 1/x
=
Intercambiando las variables: =
Wβ
= ; β 0
!β
= u
, β01, β .
, β 0 β β, β1 .
- !β
,Wβ
= ?
!β
,Wβ
= !β
,Wβ
+ !β
,Wβ
!β
,W:
123 Wβ
β© - !β
= Β’1 β 0QΒ£ β©01, β.
01, β .
123 Wβ
β - !β
β
- !β
,Wβ
= β - W β§ W β - !β
Q
= < 0 Γ³ > 0Q β§ > 1
< 0 Γ³ > 0Q β§ > 0
< 0 Γ³ > 0Q β§ β 1 < 0
< 0 Γ³ > 0Q β§ β 00,1. Q
β 00,1 .
- !β
,Wβ
= 00,1 .
!β
,W:
170. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
123 Wβ
β© - !β
= Β’1 β 0QΒ£ β©0 β β, β1.
β 0 β β, β1 .
123 Wβ
β - !β
β
- !β
,Wβ
= β - W β§ W β - !β
Q
= < 0 Γ³ > 0Q β§ < β1
< 0 Γ³ > 0Q β§ < 0
< 0 Γ³ > 0Q β§ + 1 < 0
< 0 Γ³ > 0Q β§ β 0 β 1,0.Q
β 0 β 1 0 .
- !β
,Wβ
= 0 β 1, 0.
- !β
,Wβ
= - !β
,Wβ
+ - !β
,Wβ
- !β
,Wβ
= 00,1 . s 0 β 1, 0.
- !β
,Wβ
= 0 β 1,1 . β 0Q
110)
! = 2 + G, β .β3,30, 2 <
a) β = ! + !β
=
N
+
= 2 + G β β G = 2
=
Γ
]
; @3[>)'2?G@23L, > :
171. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
=
Γ
]
β !β
=
Γ
]
De: ! + !β
=
N
+
2 + G +
Γ
]
=
N
+
2 + 2G + β G =
N
2 + 2
2 + 1 + 2G β G =
N
2 + 2
β u
2 + 1 =
N
2
2G β G = 2
2 + 1 =
N
2 β 22 β 52 + 2 = 0
2 =
NΒ±β N ^
%
=
NΒ±
%
2 = 2 ; 2 =
Como: 2 > β 2 = 2
De; 2G β G = 2
G 2 β 1 = 2
G 2 β 1 = 3
G = 3
β 2 = 2 G = 3
! = 2 + 3
b) W = | + 3| β | + 1| ; !,W = ?
172. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
< β3 ; | + 3| = β + 3
| + 1| = β + 1
W = β β 3 + + 1
W = β2
-3β€ < β1 ; | + 3| = + 3
| + 1| = β + 1
W = + 3 + + 1
W = 2 + 4
> β1 ; | + 3| = + 3
| + 1| = + 1
W = + 3 β β 1
W = 2
W = u
β2 , β 0 β β, β3.
2 + 4 , β .β3, β1.
2 , β .β1, β .
!,W = !,W + !,W + !,W
!, W :
123 W β© - f = β2Q β© 0 β β, β3.
= β
173. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
β β !, W
!, W :
β3 β€ < β1
β6 β€ 2 < β2
β2 β€ 2 + 4 < 2
123 W = .β2,2 .
123 W β© - f = .β2,2.β© .β3,30
= .β2,2 .
123 W β - !β
β
- f, W = - W
- fΓ ΓΓ = .β3, β1.
f, W = !.W 0 = ! 2 + 4
!.W 0 = 2 2 + 4 + 3
!.W 0 = 4 + 11
!, W :
123 W β© - f = 2Q.β© .β3,30
= 2Q
123 W β - !β
β
- f, W = - W
- fΓ ΓΓ = .β1, β.
f, W = !.W 0 = ! 2
!.W 0 = 2 2 + 3
174. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
!.W 0 = 7
!,W = P
4 + 11 , β .β3, β1.
7 , β .β1, β .
111)
! = u
10 β 2 , < 0
β + 16 , 0 β€ β€ 3
O %
, > 3
; W = |
β β 10 β 21 . β .β5, β10
| |
| |
, β 01, 20
! = 10 β 2
De: , β .ββ, 0 .; ! = ! β =
10 β 2 = 10 β 2
β2 = β2
= β β β β@3 >'[@t2
< 0 β 2 < 0
β2 > 0
10 β 2 > 10
123 ! = .10, β .
De: , β .0, 3 0; ! = ! β =
: + 16 = : + 16
| + 16| = | + 16|
+ 16 = + 16
175. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
=
= β β β β@3 >'[@t2
0 β€ β€ 3 β 0 β€ β€ 9
16 β€ + 16 β€ 25
4 β€ β + 16 β€ 5
123 ! = .4,50
De: , β 03, β .; ! = ! β =
β
O %
=
O
O %
β 4 = β 4
=
= β β β β@3 >'[@t2
> 3 β > 9
β 4 > 4
O %
< 4
123 ! = .ββ, 4.
Se tiene que: 123 ! β© 123 ! = β
123 ! β© 123 ! = β
123 ! β© 123 ! = β
β ! >= @3 >'[@t2
b)
W = u
β β 10 β 21 , β .β5, β10
| β 2| β 1
| + 3|
, β 01, 20
176. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
01, 2 . β | β 2| = β β 2
| + 3| = + 3
W =
| |
| |
= =
β₯ 2 β | β 2| = β 2
| + 3| = + 3
W =
| |
| |
= =
W = β’
β 2
β 10 β 21 . β .β5, β10
, β 01,2 .
, = 2
W = β β 10 β 21
, β 0 β 5, β10; ! = ! β =
β β 10 β 21 = β β 10 β 21
β β 10 = β β 10
+ 10 = + 10
β +10 ( β = 0
β + + + 10 β = 0
β + + 10 = 0
+ + 10 β 0
177. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
β β = 0
= β β β β@3 >'[@t2
W =
, β 01,2 .; ! = ! β =
β
β
= O
O
+ 3 β β 3 = β + 3 β 3
4 = 4
= β β β β@3 >'[@t2
W =
, = 2; ! = ! β =
βΒΊΕ
β
= OΒΊΕ
O
+ 3 β 3 β 9 = β 3 + 3 β 9
6 = 6
= β β β β@3 >'[@t2
Los rangos con:
123 W = .β12,40
123 W =0 β
N
, 0 .
123 W = β
N
123 W β© 123 W = 0 β 1/5 , 0 .
123 W β© 123 W β β
----------------g (x) no es inyectiva
178. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
112) Probar que si f(x) = 4β β , 0 β€ β€ 1 E,=>> !β
,
calcule su inversa.
f(x) = 4β β , 0 β€ β€ 1
, β - ! ; ! = ! β =
4β β = 4β β
4 β β β β β = 0
4 β β β β β β β + β = 0
β β β 4 + β + β = 0
Si: 0 β€ β€ 1 β 4 + β + β β 0
β β β β = 0
β = β
| | = | |
= β β β β@3 >'[@t2
113)
De: 1 <
%
%
β€ 10
1 <
%
%
β§
%
%
β€ 10
%
%
β 1 > 0 β§
%
%
β 10 β€ 0
β
6
%
> 0 β§
6 ^
%
β€ 0
179. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
6
%
< 0 β§
6 ^
%
β€ 0
4 β 2 < 0 β§ 4 β 2 9 β 36 β€ 0
Resolviendo las inecuaciones, se tiene:
β < 0 Γ³ > 2 β§ β€ 2 Γ³ β₯ 4
β 0 β β, 0 0 s .4, β .
= 0 β β, 0 0 s .4, β .
114)
! = 2 + '
! ' = 2!β
'
a) = 2 + ' β =
Ε‘
Intercambiando las variables: =
Ε‘
!β
=
Ε‘
! ' = 2' + ' = 3'
2!β
' = 2 $
Ε‘O Ε‘
&
β 3' = 2 $
Ε‘O Ε‘
&
3' = 2'
Ε‘
)
3 = ' β 1
' = 4
180. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
β ! = 2 + 4
!β
=
%
! 0 . !β
0 = 2.0 + 4 . $
` %
&
= 4 β2 = β8
! 0 . !β
0 = β8
b)
β = ?
β =
. %
βΒΊβ
O
=
^
ΒΊΕ
O
= β4
β = -4
115)
a) ! = ; β 2
! = 2 +
N
- ! = 1 β 2Q
= 0 β β, 2 . s 02, β .
De: < 2 Γ³ > 2
β 2 < 0 Γ³ β 2 > 0
< 0 Γ³ > 0
N
< 0 Γ³
N
> 0
181. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
2+
N
< 2 Γ³ 2 +
N
> 2
! < 2 Γ³ ! > 2
123 ! = 0 β β, 2 . s 02, β .
b) ! =
, β - ! ; ! = ! β =
β
β
= O
O
2 β 4 + β 2 = 2 + β 4 β 2
β4 + = β 4
5 = 5
= ---------------inyectiva
De; =
β 2 = 2 + 1
β 2 = 2 + 1
= ;
@3[>)'2?G@23L, +2= t2)@2G+>=: =
2 + 1
β 2
!β
=
116
a)
Si, < β ! > !
182. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
< β ! > !
β β β β β β β β L>')>'@>3[>
Si: x >0
< β ! > !
β β β β β β β β L>')>'@>3[>
β β β β β β β β β βL>')>'@>3[>
G de las grΓ‘ficas se aprecia que la funciΓ³n es inyectiva, por
tanto existe la inversa:
183. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
β = β 2 + 2
β = β 2 + 1 + 1
β = β 1 + 1
= β 1 + 1
123 ~β β’ = . ! 0 , β .
123 ~β β’ = .2, β .
β 1 = β 1
x-1 = Β± : β 1
= 1 Β± : β 1
= 1 + β β 1
ββ
= 1 β β β 1
β = β3 β 6 + 2
β = β3 + 2 + 1 + 5
β = 5 β 3 + 1
= 5 β 3 + 1
123 ~β β’ =0 β β , ! 0 .
123 ~β β’ =0 β β, 2 .
3 + 1 = 5 β
+ 1 =
N
β + 1 = Β± M
N
= Β± M
N
β 1
Intercambiando la variable:
= Β± M
N
β 1
Y < 2 β = M
N
β 1
184. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
ββ
= u
1 β β β 1 , β .2, β .
M
N
β 1 , β 0 β β, 2 .
117
! = | |
| | = B
, β₯ 0
β , < 0
Si, xβ₯ 0 βΆ | | =
! =
< 0 | | = β
! =
185. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
! = u
, .0,1 .
, 0 β 1,0 .
! =
, β - ! ; ! = ! β =
β
=
O
β 1 β = 1 β
= β β β β@3 >'[@t2
! =
, β - ! ; ! = ! β =
β
=
O
β 1 + = 1 +
= β β β β@3 >'[@t2
Se analiza los rangos de ! ! ,Γ
Γ {ΓΓ
Γ ΓΒ½Γ ΓΓ] β
0 β€ < 1
β1 β€ β < 0
0 β€ 1 β < 1
> 1
123 (! = 01, β .
DE; β1 < < 0
0 β€ + 1 < 1
1 <
123 (! = 01, β .
186. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
123 (! β© 123 (! β β
β 3, => [@>3> @3t>)=2
118)
Se debe demostrar que son inyectivas cada una
de las funciones que forman parte de f(x);
Graficar las funciones y analizar sus rangos, se verΓ‘ que es
inyectiva,
! = β β 2
! = 2 + β3 + 2 β
187. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
123 ! β© 123 ! = β
! β β β @3 >'[@t2
! β3 = β9 + 6 = β3
! β1 = β1 + 2 = 1
123 ! = .β3, 1.
! β1 = 2 + β3 β 2 β 1 = 2
! β1 = 2 + β3 + 2 β 1 = 4
123 ! = .2, 40
! = β β 2
= β + 2 + 1 + 1
1 β = + 1 ; xβ .β3, β1. , se toma
+2 )2@J2 3>W2[@t2 L> 1βy)
+ 1 = Β± :1 β
= Β± :1 β β 1
Intercambiando variables:
= Β± β1 β β 1
!β
= ββ1 β β 1
188. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
! = 2 + :3 + 2 β
= 2 + :β β 2 + 1 + 4
β 2 = :4 β β 1
β 2 = 4 β β 1
β 1 = 4 β β 2
β 1 = Β±:4 β β 2 ; xβ .β1,10, se
toma la raΓz negativa de (4 β β 2 :
= 1 Β± :4 β β 2
Intercambiando variables:
= 1 Β± :4 β β 2
!β
= 1 β :4 β β 2
!β
= r
ββ1 β β 1 , β .β3, 1.
1 β :4 β β 2 , β .2,40
w> L>G> ',3=@L>)2) K*>:
!: β
!β
: β
189. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
119
!: β .β9, β1. ; ! =
%
2 =?
β9 β€
%
< β1
β β9 β€
%
β§
%
< β1
β 0 β€ 9 +
%
β§
%
+ 1 < 0
7 6 %
β₯ 0 β§
%
< 0
` N
β₯ 0 β§
^
< 0
^
β₯ 0 β§ < 0
6 β 3 β β₯ 0 β§ 2 + 3 β < 0
β€ 3 Γ³ β₯ 6Q β§ { x <-2 Γ³ x > 3 }
β 0 β β, β20 s .6, β .
G
, β - ! ; ! = ! β =
% β
β
=
% O
O
9 β 3 + 12 β 4 = 9 + 12 β 3 β 4
3 β 3 + 12 β 12 = 0
= β β β β@3 >'[@t2
190. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
c)
Sobreyectiva:
=
%
β 3 β = 3 + 4
3 β 3 = 4 +
=
%
De: ! = !
%
)
! =
%$
ΕΒΏΒΊΕ
ΒΏββ
&
ΕΒΏΒΊΕ
ΒΏββ
%
=
% _
=
N
7 N
! β
----------------no es sobreyectiva
120)
Univalente β @3 >'[@t2
! = 12 β 4 +
= β 4 + 4 + 6 β 2
! = β 2 + 4
El rango de f es:
0β€ < 1 Γ³ 2 β€ β€ 3
β2 β€ β 2 < β1 Γ³ 0 β€ β 2 β€ 1
191. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
1 < β 2 β€ 4 Γ³ 0 β€ β 2 β€ 1
< β 2 β€ 2 Γ³ 0 β€ β 2 β€
4+ < 4 + β 2 β€ 6 Γ³ 4 β€ 4 + β 2 β€ + 4
6
< 4 + β 2 β€ 6 Γ³ 4 β€ 4 + β 2 β€
6
Ran (f) = 0
6
, 60 s .4,
6
0
123 ! = .4,60
De:
, β - ! ; ! = ! β =
β 2 + 4 = β 2 + 4
β 2 = β 2
: β 2 = : β 2
| β 2| = | β 2|
β 2 = β 2
= β β β β@3 >'[@t2
De: = β 2 + 4
2 = β 2 + 8
2y-8 = β 2
β 2 = Β±:2 β 8
= 2 Β± :2 β 8
Intercambiando las variables:
= 2 Β± β2 β 8
Si: β 0
6
, 60
!β
= 2 β β2 β 8
192. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
Si : β .4,
6
.
!β
= 2 + β2 β 8
! = u
2 + β2 β 8 , β .4,
6
.
2 β β2 β 8 , β 0
6
, 60
121
! = + 1
! = β + 2
! = + 1
De:
, β - ! ; ! = ! β =
+ 1 = + 1
=
= β β β @3 >'[@t2
193. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
! = β + 2
De:
, β - ! ; ! = ! β =
: + 2 = : + 2
| + 2| = | + 2|
+ 2 = + 2
= β β β @3 >'[@t2
123 ! =03,90
123 ! = .0,20
123 ! β© 123 ! = β
β ! β β β β β @3 >'[@t2
De: = + 1 β 2 = + 2
= 2 β 2 ; β .β4, β2.
< 0 β = β:2 β 2
Intercambiando las variables:
= β β2 β 2
!β
= ββ2 β 2
194. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
! = β + 2
= β + 2 ; + 2 =
= β 2
Intercambiando variables:
= β 2
!β
= β 2
!β
= r
ββ2 β 2 , β 03,90
β 2 , β .0,20
122)
! = 4 β
! =
O
De:
195. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
, β - ! ; ! = ! β =
4 β = 4 β
β β 4 β = 0
( β + β 4 β = 0
( β + β 4 = 0
β 0 β β, 2. β + β 4 β 0
( β = 0
= β β β β@3 >'[@t2
, β - ! ; ! = ! β =
β
O
β
= O
O
O
β 2 = β 2
β β 2 β = 0
β β 2 β + = 0
β β 2 β 2 = 0
β 02, 4 . β β 2 +
β = 0
= β β β β@3 >'[@t2
Los rangos de f1 y f2 son:
123 ! = 0 β β, 4.
123 ! = 0 8, β.
196. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
123 ! β© 123 ! = β
β ! β β β β@3 >'[@t2
! = 4 β
= β β 4 + 4 + 4
= 4 β β 2
β 2 = 4 β
x-2 < 0 β [,?2) +2 )2ΓJ 3>W2[@t2 L> 4 β
β 2 = β:4 β
= 2 β :4 β
Intercambiando las variables:
= 2 β β4 β
!β
= 2 β β4 β
! =
β 2
=
O
β 2 =
β +
%
=
%
β 2
β =
%
β 2
197. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
β = Β±M
%
β 2
= Β± M
%
β 2
Intercambiando las variables:
= β M
%
β 2
!β
= β M
%
β 2
!β
= u
2 β β4 β , < 4
β M
%
β 2 , > 8
123)
! = + 2 + 2
198. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
! = + 4
Realizando las grΓ‘ficas de f1 y f2 y al trazar respectivamente una
recta paralela al eje x, se observa que se corta en un solo punto,
esto implica que son inyectivas
123 ! β© 123 ! = β
! = + 2 + 2
= + 2 + 1 + 1
β 1 = + 1
+ 1 = β 1
β₯ 1 β + 1 > 0 β β β [,?2) +2 )2ΓJ E,=@[@t2 L>
y-1
+ 1 = : β 1
= : β 1 β 1
Intercambiando las variables:
= β β 1 β 1
!β
= β β 1 β 1
! = + 4
= + 4
199. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
= β 4
= : β 4
Ε
Intercambiando las variables:
= β β 4
Ε
!β
= β β 4
Ε
!β
= r
β β 1 β 1 , β₯ 5
β β 4
Ε
, < 5
124)
! = β 1
! = + 1
W = 2 β 1
W = β
Verificar si g(x) es inyectiva:
200. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
, β - W ; W = W β =
2 β 1 = 2 β 1
2 = 2
= β β β @3 >'[@t2
, β - W ; W = W β =
β = β
β = β
| | = | |
= β β β @3 >'[@t2
Los rangos de las funciones g, son:
123 W = 0 β β, β1.
123 W = 00, β.
123 W β© 123 W = β
β W β β β β@3 >'[@t2
W = 2 β 1 β = 2 β 1
=
Cambiando las variables:
201. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
=
Wβ
= + 1
W = β β = β
=
Cambiando variables: =
Wβ
=
Wβ
= r
, < β1
, β₯ 0
Grafica de g y g*
!,Wβ
= ! ,Wβ
+ ! ,Wβ
+ ! ,Wβ
+ ! ,Wβ
123 Wβ
=0 β β, 0 .
123 Wβ
= .0, β .
! ,Wβ
:
123 Wβ
β© - ! =[- β, 0 .β© 0 β β, β1. =0 β β, β1.
123 Wβ
β’ - ! β - ! ,Wβ
=
= / β - Wβ
β§ Wβ
Γ - ! }
202. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
< 0 β§ < β1
< 0 β§ + 1 < β2
< 0 β§ < β3
β 0 β β, β3.
- ! ,Wβ
= 0 β β, β3.
! .Wβ 0 = ! $ &
= $ & β 1 =
%
+ β
%
! .Wβ 0 = $ + β &
! ,Wβ
:
123 Wβ
β© - ! =[ 0, β .β© 0 β β, β1.= β
β β ! ,Wβ
! ,Wβ
:
123 Wβ
β© - ! =] ββ, 0 . β© .β1, β.= .β1,0.
123 Wβ
β’ - ! β - ! ,Wβ
=
= / β - Wβ
β§ Wβ
Γ - ! }
< 0 β§ β₯ β1
< 0 β§ + 1 β₯ β2
< 0 β§ β₯ β3
β .β3,0.
- ! ,Wβ
= . β3,0.
! .Wβ 0 = ! $ &
203. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
= + 1 =
! .Wβ 0 =
! ,Wβ
:
123 Wβ
β© - ! =[0, β . β© .β1, β.=[0,β.
123 Wβ
β’ - ! β - ! ,Wβ
=
= / β - Wβ
β§ Wβ
Γ - ! }
β₯ 0 β§ β₯ β1
β₯ 0 β§ + 1 β₯ 0
β₯ 0 β§ β 1
β .0, β .
- ! ,Wβ
= .0, β.
! .Wβ 0 = !
= + 1
! .Wβ 0 = + 1
!,Wβ
=
β©
β¨
β§ $ + β & , < β3
, β .β3,0.
+ 1 , β .0, β .