1. 9
(Increasing Function) T.KAINOY
(Decreasing Function)
(Increasing Function) x2 > x1
f (x2) > f (x1)
x f (x)
f(x) = 2x – 6
(Decreasing Function) x2 > x1
f (x2) < f (x1)
x f (x)
f(x) = -3x+5
1. f + g = {(x , y) y = f(x) + g(x)} Df+g =
Df Dg
2. 10
T.KAINOY
2. f - g = {(x , y) y = f(x) - g(x)} Df-g = Df Dg
3. f . g = {(x , y) y = f(x).g(x)} Dfg = Df Dg
4. f
= {(x , y) y = f (x )
} D f
= Df
g g (x ) g
Dg-{x g(x) = 0}
9
f + g , f – g , fg
(1)
(2)
(3)
(4)
( )
10 f ( x) 2 x 1 g ( x) 3 x 4
f
(f g )(x) , (f g )(x) , ( fg)(x) (x)
g
(1) (f g )(x) = f ( x) g ( x)
= (2 x 1) (3x 4)
= ………………………………
(2) (f g )(x) = f ( x) g ( x)
= (2 x 1) (3x 4)
3. 11
T.KAINOY
= ………………………………
(3) ( fg)(x) = f ( x ) g ( x)
= (2 x 1)(3x 4)
=
…………………………………………………………………………
=
…………………………………………………………………………
f f ( x)
(4) ( )( x ) =
g g ( x)
=
…………………………………………………………………………
(Composite
function)
f g
f g
4. 12
T.KAINOY
f g gof
g f fog
g f
fog
gof f
g
fog g
gof(x) = g(f(x))
f
fog(x) = f(g(x))
11
gof
6. 14
T.KAINOY
= f(x2-1)
= 2(x2-1) + 5
= 2x2 + 3
14 gof(3)
fog(1)
Rf …….. Dg …….
Rf Dg ≠ gof
gof(x) = g(f(x))
=
=
=
=
gof(3) =
Rg …… Df …….
Rg Df ≠ fog
fog(x) = f(g(x))
=
=
=
fog(1) =
1) gof fog gof
f g fog
g f
2) Dgof = Df Rgof Rg
7. 15
T.KAINOY
3) (fog)oh = fo(goh)
4) gof f g
(Inverse
Function)
f-1
f-1 = (y,x) (x,y) f
r
f
f f −1 f −1
f –1
f − 1(x)
= (0,3) ,(1,4) ,(2,5)
= (3,0) ,(4,1) ,(5,2)
= (0,3) ,(1,3) ,(2,4)
= (3,0) ,(3,1) ,(4,2)
9. 17
T.KAINOY
3) ( gof ) 1 ( x) f 1 ( x)og 1 ( x)
4) ( fof 1
)( x) ( f 1of )( x)
2
1 (x,y)
f ( x, y) A B| y 2x 7
1
f ( y, x) B A | y 2x 7
2 x y
f ( x, y) A B| y 2x 7
1
f ( x, y ) B A | x 2y 7
y =
1 x 7
f ( x, y ) B A| y
2
17 f ( x, y ) A B| y
3x 1
2x 5
………………………………………………………………………………
………………………………………………………………………………
…………………………………………………………………………….
………………………………………………………………………………
………………………………………………………………………………
…………………………………………………………………………….
10. 18
T.KAINOY
………………………………………………………………………………
………………………………………………………………………………
…………………………………………………………………………….
………………………………………………………………………………
………………………………………
18 f (x) (1,5), (2,4), (3,6), (7,9)
g (x) (4,1), (2,2), (1,4), ( 2,7)
1
(f g )( x)
f (x) (1,5), (2,4), (3,6), (7,9)
g (x) (4,1), (2,2), (1,4), ( 2,7)
g 1 ( x) (1,4), (2,2), (4,1), (7, 2)
Df 1,2,3,7 Dg 1 1,2,4,7
Df Dg 1 1,2,7
(f g 1 )( x) = f ( x) g 1 ( x)
= (1,......... .), (2,......... ..), (7,......... ....)
= ………………………………………..
19 f ( x) 2x 3 g ( x) x2 5
1 1
f g (5)
f (x) y 2x 3
1
f ( x) x 2y 3
x 3
y
2
2
g (x) y x 5
g 1 ( x) x y2 5
y x 5
11. 19
T.KAINOY
f 1
g 1
( x) = f 1
( x) g 1 ( x)
= x 3
x 5
2
f 1
g 1
(5) =
………………………………………………………………
=
………………………………………………………
………
20 f ( x) 3 x 5 g ( x) x2 1
f 1og 1 ( x)
f (x) y 3x 5
1
f ( x) x 3y 5
x 5
y
3
2
g (x) y x 1
g 1 ( x) x y2 1
y x 1
f 1og 1 ( x) = f 1 ( g 1 ( x))
= f 1
( x 1)
= x 1 5
3
12. 20
T.KAINOY
21 f (x) (0,5), (1,9), (2,10 ), (3,15)
g (x) (3,5), ( 1,9), (2,10 ), (4,7)
g 1of ( x)
g 1 ( x) (5,3), (9, 1), (10 ,2), (7,4)
g 1of ( x) =………………………………………………………..
1. f g f 1 ( x)
x 4
( fog )( x) 3 x 2 2
3
f ( x) g ( x)
x 4 x 4
f 1 ( x) y
3 3
f y x x
y
y 4
x
3
y 3x 4
f ( x) 3 x 4
( fog )( x) 3 x 2 2
f ( g ( x)) 3x 2 2
3g ( x) 4 3 x 2 2
g (x) ………………………………..
13. 21
T.KAINOY
g (x) ………………………………..
f ( x) g ( x) =
……………………………………………………………………
…………..
=
……………………………………………………………………
…………..
2. f ( x 1) 3x 2 f ( x) g (3x 1) 2 x 8
f (0) 1 g 1 ( f ( 2))
f ( x 1) 3x 2 f ( x) f( ) = f-
x 0 f (0 1) 3(0) 2 f (0) 1( ) =
f (1) 0 2 1 f (1) 3
x 1 f (1 1) 3(1) 2 f (1)
f (2) 3 2 3 f (2) 8
g 1 ( f (2)) g 1 (8)
g (3x 1) 2 x 8
g 1 (2 x 8) 3x 1
2x 8 8 x 0
g 1 (2 x 8) g 1 (8) 3(0) 1 1
g 1 ( f (2)) 1
3. f ( x) x2 5 g ( x) x 5
(1) (gof)(5) (2) (fog)(7)
(1) (gof)(5) =
………………………………………………………………………………
……..
