This document provides an overview of functions and key concepts in calculus and analytic geometry. It defines what a function is, including the domain and range. It describes different types of functions such as polynomial, linear, identity, constant, rational, exponential, and logarithmic functions. Examples are given for each type of function. Key aspects like the vertical line test and graphs of functions are also summarized.

1. mathcity.org
Merging man and maths
Exercise 1.1
Calculus and Analytic Geometry, MATHEMATICS 12
Available online @ http://www.mathcity.org, Version: 2.0.0
Function and Limits
Concept of Functions:
Historically, the term function was first used by German
mathematician Leibnitz (1646-1716) in 1673 to denote the dependence of one quantity on
another e.g.
1) The area “A” of a square of side “x” is given by the formula A=x2
.
As area depends on its side x, so we say that A is a function of x.
2) The area “A” of a circular disc of radius “r” is given by the formula
A=π r2
As area depends on its radius r, so we say that A is a function of r.
3) The volume “V” of a sphere of radius “r” is given by the formula
V= 3
3
4
rπ . As volume V of a sphere depends on its radius r, so we say that
V is a function of r.
The Swiss mathematician, Leonard Euler conceived the idea of denoting function
written as y=f(x) and read as y is equal to f of x. f(x) is called the value of f at x or image
of x under f .
The variable x is called independent variable and the variable y is called
dependent variable of f.
If x and y are real numbers then f is called real valued function of real numbers.
Domain of the function:
If the independent variable of a function is restricted to lie
in some set, then this set is called the domain of the function e.g.
Dom of f = {0 ≤ x≤ 5}
Range of the function:
The set of all possible values of f(x) as x varies over the
domain of f is called the range of f e.g. y = 100 – 4x2
.
As x varies over the domain [0,5] the values of y = 100 – 4x2
vary between y=0 (when
x=5) and y = 100 (when x=0)
Range of f = {0 ≤ y ≤ 100}
Definition:
A function is a rule by which we relate two sets A and B (say) in such a
way that each element of A is assigned with one and only one element of B. For example
is a function from A to B.
x
x
x
x
r

2. Ex # 1.1 – FSc Part 2
2
its Domain = {1,2,3} and Range = {4,5}
1
2
3
4
5
In general:
A function f from a set ‘X’ to a set ‘Y’ is a rule that assigns to each
element x in X one and only one element y in Y.(a unique element y in Y)
x y=f(x)
f
X Y
(f is function from X to Y)
If an element “y, of Y is associated with an element “x, of X, then we write y=f (x) &read
as y” is equal to f of x. Here f(x) is called image of f at x or value of f at x .
Or if a quantity y depends on a quantity x in such a way that each value of x determines
exactly one value of y. Then we say that y is a function of x.
The set x is called Domain of f . The set of corresponding elements y in y is called
Range of f . we say that y is a function of x.
Exercise 1.1
Q1. (a) Given that f(x) = x2
– x
i. f(-2) = (-2)2
– (-2) = 4 + 2 =6
ii. f(0) = (0)2
– (0) = 0
iii. f(x-1) = (x-1)2
– (x-1) = x2
– 2x + 1 – x + 1 = x2
– 3x + 2
iv. f(x2
+4) = (x2
+4)2
- (x2
+4) = x4
+ 8x2
+ 16 – x2
– 4 = x4
+ 7x2
+ 12
(b) Given that 4)( += xxf

3. Ex # 1.1 – FSc Part 2
3
2 2 2
) ( 2) 2 4 2
) (0) 0 4 4 2
) ( 1) 1 4 3
) ( 4) 4 4 8
i f
ii f
iii f x x x
iv f x x x
− = − + =
= + = =
− = − + = +
+ = + + = +
( )
2.
) ( ) 6 9
( ) 6( ) 9 6 6 9
( ) 6 9
( ) ( ) (6 6 9) (6 9)
6 6 9 6 9 6
6
) ( ) sin
sin sin 2cos sin
2 2
( ) sin( ) sin
( ) (
Q Giventhat
i f x x
f a h a h a h
f a a
f a h f a a h a
Now
h h
a h a h
h h
ii f x x given
f a h a h and f a a
f a h f
Now
θ ϕ θ ϕ
θ ϕ
= −
+ = + − = + −
= −
+ − + − − −
=
+ − − +
= = =
=
+ +   
− =    
   
+ = + =
+ −
Q
) sin( ) sina a h a
h h
+ −
=
[ ]
1
sin( ) sin
1 1 2
2cos sin 2cos sin
2 2 2 2
1 2 2
2cos sin cos sin
2 2 2 2 2
a h a
h
a h a a h a a h h
h h
a h h h h
a
h h
= + −
 + + + −   +        
= =          
          
        
= + = +        
        

4. Ex # 1.1 – FSc Part 2
4
3 2
3 2 3 3 2 2
3 3 2 2 2 2
3 2
3 3 2 2 2 2 3 2
3 3 2 2 2
) ( ) 2 1
( ) ( ) 2( ) 1 3 ( ) 2( 2 ) 1
3 3 2 4 2 1
( ) 2 1
( ) ( )
3 3 2 4 2 1 ( 2 1)
1
3 3 2
iii Giventhat f x x x
f a h a h a h a h ah a h a ah h
a h a h ah a ah h
f a a a
Now f a h f a
a h a h ah a ah h a a
h
a h a h ah a
h
= + −
+ = + + + − = + + + + + + −
= + + + + + + −
= + −
+ −
+ + + + + + − − + −
=
= + + + + 2 3 2
3 2 2 2 2 2
2 2 2 2 2 2
4 2 1 2 1
1
3 3 4 2 3 3 4 2
3 3 4 2 3 2 3 4 (3 2) 3 4
ah h a a
h
h a h ah ah h h a ah a h
h h
h a ah a h h ah h a a h a h a a
 + + − − − + 
   = + + + + = + + + +   
= + + + + = + + + + = + + + +












+
−
=
















 +
−=
−+
=
−+
=
+=+
=
2
sin
2
sin
2
2
sin
2
2
sin2
1cos)cos(
)()(
cos)(
)cos()(
cos)()
hh
a
h
hha
hh
aha
h
afhaf
Now
aafand
hahafso
xxfthatGiveniv
Q3. (a) If ‘x’ unit be the side of square.
Then its perimeter P = x+ x+ x + x = 4x ……………….. (1)
A = Area = x . x = x2
…………… (2)
From (2) Ax = putting in (1)
P = 4 A
∴ P is expressed as Area
(b) Let x units be the radius of circle
Then Area = A = 2
xπ ……………….. (1)
Circumference = C = xπ2 ……………….. (2)
From (2) x =
π2
C
Putting in (1)
A =
ππ
π
π
π
442
2
2
22
ccc
=





=





A = nceCircumfereoffunctionaisArea
c
Q
π4
2
(c) Let x unit be each side of cube.
The Volume of Cube = x . x . x = x3
……………….. (1)
Area of base = A = x2
……………….. (2)
From (2) x = A Putting in (1)
x
x
x
x
x
x x
x

5. Ex # 1.1 – FSc Part 2
5
V = ( ) 2
33
)(AA =
1)(.5 23
++−= bxaxxxfQ
( )
2)2(2
063
0
062
)2()1(
1.................026
2
)2(.................002412
03249
01131248
01)1()1()1(31)2()2()2(
0)1(3)2(
2323
−=⇒−=⇒=
=−
=+
=−−
=−+−
−
=+=+−
=−−−=+−
=+−−−−=++−
=+−+−−−−=++−
=−−=
babanda
a
ba
ba
andSolving
ba
byDividing
baba
baba
baba
aba
fandfIf
m
h
xa
xxhQ
30
)1(1040)1(
sec1)(
1040)(.6
2
2
=
−=
=
−=
m
h
xb
5.175.2240)25.2(1040
)5.1(1040)5.1(
sec5.1)(
2
=−=−=
−=
=
m
h
xc
1.119.2840)89.2(1040
)7.1(1040)7.1(
sec7.1)(
2
=−=−=
−=
=
ii) Does the stone strike the ground = ?
2
44010
01040
0)(
22
2
±=
=⇒−=−
=−
=
x
xx
x
xh
Stone strike the ground after 2 sec.
Graphs of Function
Definition:

6. Ex # 1.1 – FSc Part 2
6
The graph of a function f is the graph of the equation y = f(x). It consists
of the points in the Cartesian plane chose co-ordinates (x , y) are input - output pairs for f
.
Note that not every curve we draw in the graph of a function. A function f can have only
one value f(x) for each x in its domain.
Vertical Line Test
No vertical line can intersect the graph of a function more than once. Thus, a circle
cannot be the graph of a function. Since some vertical lines intersect the circle Twice. If
‘a’ is the domain of the function f , then the vertical line x = a will intersect the graph of f
in the single point (a , f(a)).
Types of Function
ALGEBRAIC FUNCTIONS
Those functions which are defined by algebraic expressions.
1) Polynomial Functions:
01
1
1 .............)( axaxaxaxP n
n
n
n ++++= −
− Is a
Polynomial Function for all x where 210 ,, aaa ……. na are real numbers, and
exponents are non-negative integer . na is called leading coefft of p(x) of degree n,
Where 0≠na
4deg1232)(
max
34
=−+−=
⇒
reexxxxP
equationinxofpowerimumtheisfunctionpolynomialofDegree
2) Linear Function: if the degree of polynomial fn is ‘1, is called linear function
.i.e. p(x)=ax+b
or⇒ Degree of polynomial function is one.
bxy
abaxxf
+=
≠+=
5
0)(
Q
3) Identity Function: For any set X, a function I: xX → of the form y = x or
f(x) = x. Domain and range of I is x. Note. I (x)= ax +b be a linear fn if a=1,b=0 then
I(x)=x or y=x is called identity fn
4) Constant Function: or
bydefinedyXC →: yXf →: If f (x)=c, (const) then f is
called constant fn
eg y=5
RxyorxC
RRCge
yaandXxaxC
∈∀==
→
∈∈∀=
22)(
:..
)(

7. Ex # 1.1 – FSc Part 2
7
5) Rational Function:
)(
)(
)(
xQ
xP
xR =
125
143
)(..
0)()()(
23
2
++
++
=
≠
xx
xx
xRge
xQandpolynomialarexQandxPBoth
Domain of rational function is the set of all real numbers for which Q(x) 0≠
6) Exponential Function:
A function in which the variable appears as
exponent (power) is called an exponential function.
) 0
) 2.178
) 2
exp .
x
x
x xh
i y a x R a
ii y e x R and e
iii y or y e
aresome onential functions
= ∴ ∈ >
= ∴ ∈ =
= =
7) Logarithmic Function:
x
a
y
ythenaxIf log== x >0
loglnlog
718.2)
log10)
''log
''
10
10
naturalcalledisxy
ebaseIfii
xofLogarithmcommoncalledis
ythenbaseIfi
abaseoffunctioncLogarithmiisyThen
functionicLogarithemofbasethecalledisa
aa
x
e
x
x
a
==
==
==
=
≠>Q
8) Hyperbolic Function:
We define as
2
)sinh()
xx
ee
xyi
−
−
== Sine hyperbolic function or hyperbolic sine function
x
x
xyiv
x
x
ee
ee
Tanhxyiii
yRxfunctioninehyperboliccalledis
ee
xyii
RyyRangeandRxxDom
xx
xx
xx
sinh
cosh
coth)
cosh
sinh
)
),1[,cos
2
)cosh()
}/{}/{
===
+
−
==
∞∈∈⇒
+
==
∈=∈=
−
−
−
Rx
eex
hxyv xx
∈
+
=== −
2
cosh
1
sec)
RxxDom
eex
echxyvi xx
∈≠=
−
=== −
:0{
2
sinh
1
cos)
9) Inverse Hyperbolic Function: (Study in B.Sc level)

8. Ex # 1.1 – FSc Part 2
8
( )
( )
0
11
lncos)
1
1
1
2
1
coth)
10
11
lnsec)
11
1
1
ln
2
1
)
11lncosh)
1lnsinh)
2
1
1
2
1
1
21
21
≠







 +
+==
>
−
+
==
≤<







 −
+==
<≠
−
+
==
>∈∀−+==
∈∀++==
−
−
−
−
−
−
x
x
x
x
xechyvi
x
x
x
xyv
x
x
x
x
xhyiv
xandx
x
x
xTanhyiii
xandRxforxxxyii
Rxforxxxyi
Q
10) Trigonometric Function:
Functions Domain(x) Range(y)
) sin 1 1
) cos 1 1
i y x All real numbers y
x
ii y x All real numbers y
x
= − ≤ ≤
− ∞ < + ∞
= − ≤ ≤
−∞ < < ∞
Q
Q
) tan (2 1) ' '
2
) cot
) sec (2 1) ( 1,1)
2
( 1 1)
) cos ( ) ( 1 1)
iii y x x R k R all real numbers
k Z
iv y x x R k R
k Z
v y x x R k R
k Z or R y
vi y ecx x R k R y
k Z
π
π
π
π
= ∈ − +
∈
= ∈ −
∈
= ∈ − + − −
∈ − − < <
= ∈ − − − < <
∈
Q
11) Inverse Trigonometric Functions:
Function Dom(x) Range(y)
π
ππ
≤≤≤≤−=⇔=
≤≤−≤≤−=⇔=
−
−
yxyxxy
yxyxxy
011coscos
22
11sinsin
1
1

9. Ex # 1.1 – FSc Part 2
9
{ }
π
ππ
π
π
ππ
<<∈=⇔=
−



−∈−−∈=⇔=






−∈−−∈=⇔=
∞<<∞−
≤≤−∈=⇔=
−
−
−
−
yRxyxxCoty
yRxecyxxCoy
yRxyxxSecy
xor
yRxTanyxxTany
0cot
0
2
,
2
)1,1(cossec
2
],0[)1,1(sec
22
1
1
1
1
12) Explicit Function:
If y is easily expressed in terms of x, then y is called an
explicit function of x.
.1..)( 3
etcxxygexfy ++==⇒
13) Implicit Function:
If x and y are so mixed up and y cannot be expressed in
term of the independent variable x, Then y is called an implicit function of x. It can be
written as. f(x , y) = 0
e.g. x2
+ xy + y2
= 2 etc.
14) Parametric Function:
For a function y =f (x) if both x& y are expressed in
another variable say ‘t’ or θ which is called a parameter of the given curve.
Such as:
i) x = at2
Parametric parabola
y = 2at
y2
= 4 a
) sec
tan
vi x a Parametricequationof hyperbola
y b
θ
θ
=
=
12
2
2
2
=−
b
y
a
x
2 2 2
) cos
sin
ii x a t Parametricequationof circle
y a t
x y a
=
=
+ =
2 2
2 2
) cos
sin
1
iii x a Parametricequationof Ellipse
y b
x y
a b
θ
θ
=
=
+ =

10. Ex # 1.1 – FSc Part 2
10
Exercise 1.1
2
2
2 2 2
2
2
7. 4 ...........................(1)
.......................( )
2 .......................( )
lim ' ' ( ) ( )
2
2 44
4
Q Parabola y ax
x at i
y at ii
y
Toe inating t from ii t putting i
a
y y y
x a x a x
a aa
y ax whichis s
⇒ =
=
=
=
  
= ⇒ = ⇒ =  
   
⇒ = (1)
.
ameas
whichisequationof parabola
2 2
) cos , sin
cos ...........( ) sin ...............( ) lim ( ) ( )
( ) ( )
1
) sec , tan
sec ............
ii x a y b
x y
i and ii Toe inating from i and ii
a b
Squaring and adding i and ii
x y
represent a Ellipse
a b
iii x a y b
x
a
θ θ
θ θ θ
θ θ
θ
= =
⇒ = =
  
+ =  
   
= =
=
2 2 2 2 2 2
2 2 2 2
2 2 2 2
....( ) tan ...................( )
( ) ( )
sec tan 1 tan tan 1
y
i ii
b
Squaring and Subtracting i and ii
x y x y x y
a b a b a b
θ
θ θ θ θ
=
   
− = − ⇒ − = + − ⇒ − =   
   
Which is equation of hyperbola
2 2 2 2
8. ( ) sinh 2 2sinh cosh
. . 2sinh cosh 2 2
2 2 4 2
sinh 2 . .
x x x x x x x x
Q i x x x
e e e e e e e e
R H S x x
x L H S
− − − −
=
    − + − −
= = = =    
    
= =
( ) ( )
2 2
2 2
2
2 2
2 2 2 2
2 2 2 2
) sec 1 tan
2
. . . 1 tan 1 1
2
2 2 4 1
2
2
x x x x
x x x x
x x x x
x x x xx x
ii hx hx
e e e e
R H S hx
e e e e
e e e e
e e e ee e
− −
− −
− −
− −−
= −
   − + −
= − = − = −   
+ + +   
+ + − − +
= = =
+ + ++

11. Ex # 1.1 – FSc Part 2
11
( ) ( )
( )
( ) ( )
( )
( ) ( ) ( )
2
2
2 2
2 22 2 2 2 2
2
2 2
2 2 2 2
2 2 2 2
1
sec . .
cosh
) cos coth 1
2 2
. . coth 1 1
2 2 4 1 1
cos 2 . .
sinh
2
x x x x x x x xx x
x x x x x x
x x x x
x xx x x x
h x L H S
x
iii eh x x
e e e e e e e ee e
R H S x
e e e e e e
e e e e
ech x L H S
xe ee e e e
− − − −−
− − −
− −
−− −
= = =
= −
+ − − + + − + − +
= − = − = = 
−  − −
+ + − − +
= = = = = =
−− −
( ) ( ) ( ) ( )
( )
3
3 3 3
3
9. ( )Q f x x x
replace xby x
f x x x x x x x f x
f x x x isodd function
= +
−
 − = − + − = − − = − + = − 
⇒ = +
( ) ( )
( ) ( ) ( )
( ) ( )
2
2
2
) 2
2
2
ii f x x
replace xby x
f x x f x
f x x isneither even nor odd
= +
−
− = − + ≠ ±
= +
( )
( ) ( ) ( ) ( ) ( )
2
2 2
) 5
5 5 .
iii f x x x
replace xby x
f x x x x x f x f x isodd function
= +
−
 − = − − + = − + = −
 
( )
( )
( )
( )
( )
( )
1
)
1
11 1
1 1 1
.
x
iv f x
x
replace xby x
xx x
f x f x
x x x
f x is neither evennor odd function
−
=
+
−
− +− − +
− = = = ≠ ±
− + − − −
( )
( ) ( ) ( ) ( )
( )
2
3
1
2 3 2
3 3
2
) 6
6 6 6
.
v f x x
replace xby x
f x x x x f x
f x is aneven function
= +
−
 − = − + = − + = + =
 
_______________________________________________________________________________________