Functions

&
Types of Functions
 Concept of Function
Area

The area “𝑨” of a square depends on one of its sides “𝒙” by

the formula 𝑨 = 𝒙𝟐, so we say that 𝐴 is a function of 𝒙.

Volume

The volume “𝑽” of a sphere depends on its radius “𝒓” by the

4
formula 𝑉 = 𝜋𝑟2 we say that 𝑉 is a function of 𝑟.
3
 Function
𝑋 𝑌
A Function 𝑓 from a set 𝑋 to a set 𝑌 is a rule or a correspondence
𝑓
that assigns to each element 𝑥 in 𝑋 a unique element 𝑦 in 𝑌.

➢ The set 𝑋 is called the domain of 𝑓.

➢ The set of corresponding elements 𝑦 in 𝑌 is called the range of 𝑓

Notation & Value of a Function

If a variable 𝒚 depends on a variable 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 variable 𝒙 is called the independent variable

of 𝒇, and the variable 𝒚 is called the dependent

variable of 𝒇.
 Function
Given 𝒇 𝒙 = 𝒙𝟑 − 𝟐𝒙𝟐 + 𝟒𝒙 − 𝟏 then find
𝟏
𝒊 𝒇 𝟎 𝒊𝒊 𝒇 𝟏 𝒊𝒊𝒊 𝒇 −𝟐 𝒊𝒗 𝒇 𝟏 + 𝒙 𝒗 𝒇 ;𝒙 ≠ 𝟎
𝒙
Solution:

𝒊 𝒇 𝟎 = 𝟎 𝟑 −𝟐 𝟎 𝟐 + 𝟒 𝟎 − 𝟏 = 𝟎 − 𝟎 + 𝟎 − 𝟏 = −𝟏

𝒊𝒊 𝒇 𝟏 = 𝟏 𝟑 −𝟐 𝟏 𝟐 +𝟒 𝟏 −𝟏=𝟏−𝟐+𝟒−𝟏=𝟐

𝒊𝒊𝒊 𝒇 −𝟐 = −𝟐 𝟑 − 𝟐 −𝟐 𝟐 + 𝟒 −𝟐 − 𝟏 = −𝟖 − 𝟖 − 𝟖 − 𝟏 = −𝟐𝟓
Function

𝒊𝒗 𝒇 𝟏 + 𝒙 = 𝟏 + 𝒙 𝟑 −𝟐 𝟏+𝒙 𝟐 +𝟒 𝟏+𝒙 −𝟏
= 𝟏 + 𝟑𝒙 + 𝟑𝒙𝟐 + 𝒙𝟑 − 𝟐 − 𝟒𝒙 − 𝟐𝒙𝟐 + 𝟒 + 𝟒𝒙 − 𝟏
= 𝒙𝟑 + 𝒙𝟐 + 𝟑𝒙 + 𝟐

𝟏 𝟏 𝟑−𝟐
𝟏 𝟐+𝟒
𝟏 𝟏 𝟐 𝟒
𝒗 𝒇 = − 𝟏 == 𝟑 − 𝟐 + − 𝟏
𝒙 𝒙 𝒙 𝒙 𝒙 𝒙 𝒙
 Function

Let 𝒇 𝒙 = 𝒙𝟐 then find domain and range of 𝒇

Solution:
𝒇(𝒙) is defined for every real number 𝒙. Further for every real
number 𝒙, 𝒇(𝒙) = 𝒙𝟐 is a non-negative real number. So
Domain of 𝒇 = ℝ
Range of 𝒇 = Set of all non-negative real numbers = [𝟎, ∞)
Function
𝒙
Let 𝒇 𝒙 = then find domain and range of 𝒇
𝒙𝟐−𝟒
Solution:
𝒙
At 𝒙 = 𝟐 and 𝒙 = −𝟐, 𝒇 𝒙 = is not defined, So,
𝒙𝟐−𝟒

Domain of 𝒇 = ℝ − {−𝟐, 𝟐}
Range of 𝒇 = Set of all real numbers
 Function

Let 𝒇 𝒙 = 𝒙𝟐 − 𝟗 then find domain and range of 𝒇

Solution:
We see that if 𝒙 is in the interval −𝟑 < 𝒙 < 𝟑, a square root of a
negative number is obtained. Hence no real number 𝑦 = 𝑥2 − 9 exists.
So,
Domain of 𝒇 = 𝒙 ∈ ℝ ∶ 𝒙 ≥ 𝟑 = −∞, −𝟑 ∪ [𝟑, ∞)
Range of 𝒇 = Set of all non-negative real numbers = [𝟎, ∞)
Algebraic Function
Algebraic functions are those functions which are 𝟐
𝒇 𝒙 = 𝟑𝒙 + 𝟓𝒙 + 𝟐
defined by algebraic expressions.

Piece-wise Function

A piece-wise function is a function which is defined

into two or more pieces

OR

by two or more rules, each defined for a different

interval.
 Graphs of Algebraic Functions

If 𝒇 is a real-valued function of real numbers, then the graph of

f in the 𝑿𝒀-plane is defined to be the graph of the equation
𝒚 = 𝒇(𝒙)
Method to draw the graph:
To draw the graph of 𝒚 = 𝒇(𝒙), we give arbitrary values of our
choice to x and find the corresponding values of 𝒚. In this way
we get ordered pairs (𝒙𝟏 , 𝒚𝟏 ) , (𝒙𝟐 , 𝒚𝟐 ), (𝒙𝟑 , 𝒚𝟑 ) etc. These
ordered pairs represent points of the graph in the Cartesian
plane. We add these points and join them together to get the
graph of the function.
Graph of Algebraic Function

Find the domain and range of the function 𝒇(𝒙) = 𝒙𝟐 + 𝟏 and draw its
graph.

Solution:
Here 𝒚 = 𝒇(𝒙) = 𝒙𝟐 + 𝟏

Domain 𝑓 = ℝ =set of all real numbers

Range 𝑓 = set of all non-negative real numbers except the points
0≤𝑦 < 1
Graph of Algebraic Function
For graph of 𝒇(𝒙) = 𝒙𝟐 + 𝟏, we assign some values to 𝒙 from
its domain and find the corresponding values in the range 𝒇
as shown in the table:
Graph of Piece-wise Function

Find the domain and range of the function

𝒙 𝒘𝒉𝒆𝒏 𝟎 ≤ 𝒙 ≤ 𝟏
𝒇(𝒙) = ቊ Also draw its graph.
𝒙 − 𝟏 𝒘𝒉𝒆𝒏 𝟏 < 𝒙 ≤ 𝟐

Solution:

Domain of 𝑓 = 0, 1 ∪ (1,2] = [0, 2]

Range of 𝑓 = 0, 1 ∪ 0,1 = [0, 1]

Graph of Piece-wise Function

Plotting the points (𝒙, 𝒚) and joining them we get two straight lines as
shown in the figure. This is the graph of the given function
 Types of Function
Classification of Algebraic of Function

(i) Polynomial Function

A function 𝑷 of the form

𝑷 𝒙 = 𝒂𝒏 𝒙𝒏 + 𝒂𝒏−𝟏 𝒙𝒏−𝟏 + 𝒂𝒏−𝟐 𝒙𝒏−𝟐 … . + 𝒂𝟐 𝒙𝟐 + 𝒂𝟏 𝒙 + 𝒂𝟎 for all 𝑥,

where the coefficient 𝒂𝒏 , 𝒂𝒏−𝟏 , 𝒂𝒏−𝟐 , … . , 𝒂𝟐 , 𝒂𝟏 , 𝒂𝟎 are real numbers 𝒇 𝒙 = 𝟑𝒙𝟐 + 𝟒𝒙 + 𝟐

and the exponents are non-negative integers, is called a Domain & Range of 𝒇 = ℝ

polynomial function.
 Classification of Algebraic of Function
𝒇 𝒙 = 𝟑𝒙 + 𝟓
(ii) Linear Function
or 𝐲 = 𝟑𝒙 + 𝟓
If the degree of a polynomial function is 𝟏, then it
Domain & Range of 𝒇 = ℝ
is called a linear function.

A linear function is of the form:

𝒇(𝒙) = 𝒂𝒙 + 𝒃 (𝒂 ≠ 𝟎), 𝒂, 𝒃 are real numbers.

 Classification of Algebraic of Function 𝒇 𝒙 =𝒙
𝐲=𝒙
(iii) Identity Function
Domain & Range of 𝒇 = ℝ
For any set 𝑿, a function 𝑰 ∶ 𝑿 → 𝑿 of the 𝑓𝑜𝑟𝑚

𝐼 𝑥 = 𝑥 ∀ 𝑥 ∈ X , is called an identity function.

Its domain and range is the set 𝑿 itself.

In particular, if 𝑿 = ℝ , then 𝑰 𝒙 = 𝒙 ∀ 𝒙 ∈ ℝ is the

identity function.
 𝑪: ℝ → ℝ

Classification of Algebraic of Function 𝑪 𝒙 =𝟑 ∀𝒙∈ℝ

Domain of 𝐂 = ℝ
(iii) Constant function
Range of 𝐂 = 𝟑
Let 𝑿 and 𝒀 be sets of real numbers. A function

𝑪 ∶ 𝑿 → 𝒀 defined by 𝑪 𝒙 = 𝒂 , ∀ 𝒙 ∈ X , 𝒂 ∈ 𝒀 & fixed,

is called a constant function.

 Classification of Algebraic of Function

(v) Rational Function

𝑷 𝒙
A function 𝑹(𝒙) of the form , where both 𝑷(𝒙) and
𝑸 𝒙

𝑸(𝒙) are polynomial functions and 𝑸(𝒙) ≠ 𝟎, is called a

rational function. The domain of a rational function

𝑹(𝒙) is the set of all real numbers 𝒙 for which 𝑸(𝒙) ≠ 𝟎.

 Trigonometric Function
We denote and define
trigonometric functions as follows:
 Inverse Trigonometric
Function

We denote and define inverse

trigonometric functions as follows:
Exponential Function

A function, in which the variable appears

as exponent (power), is called an

exponential function.

Example:
Hyperbolic Function
Inverse Hyperbolic Function
The inverse hyperbolic functions are expressed in
terms of natural logarithms
Explicit Function
If 𝒚 is easily expressed in terms of the
independent variable 𝒙, then 𝒚 is called an
explicit function of 𝒙.

Implicit Function

If 𝒙 and y are so mixed up and 𝒚 cannot be

expressed in terms of the independent variable 𝒙,
then 𝒚 is called an implicit function of 𝒙.
 Parametric Function

Sometimes, a curve is described by

expressing both 𝒙 and y as function of a

third variable “𝒕” or “𝜽" which is called a

parameter. The equations of the type

𝒙 = 𝒇(𝒕) and 𝒚 = 𝒈(𝒕) are called the

parametric equations of the curve .

Even Function

A function 𝒇 is said to be even if

𝒇(−𝒙) = 𝒇(𝒙) ,
𝒇 𝒙 = 𝒙𝟐
for every number 𝒙 in the domain of 𝒇.
𝒇 −𝒙 = −𝒙 𝟐 = 𝒙𝟐

⟹ 𝒇 𝒙 𝒊𝒔 𝒆𝒗𝒆𝒏
Odd Function

A function 𝒇 is said to be odd if

𝒇 −𝒙 = −𝒇(𝒙) ,
𝒇 𝒙 = 𝒙𝟑
for every number 𝒙 in the domain of 𝒇.

𝒇 −𝒙 = −𝒙 𝟑 = −𝒙𝟑

⟹ 𝒇 𝒙 𝒊𝒔 𝒐𝒅𝒅
Determine whether the following functions are even or odd
𝟑𝒙
(a) 𝒇(𝒙) = 𝟑𝒙⁴ − 𝟐𝒙² + 𝟕 (b) 𝒇 𝒙 =
𝒙2 + 𝟏

(c) 𝒇(𝒙) = sin 𝒙 + cos 𝒙

Solution:
(a) 𝒇 −𝒙 = 𝟑 −𝒙 4
− 𝟐 −𝒙 2
+ 𝟕
= 𝟑𝒙⁴ − 𝟐𝒙² + 𝟕 = 𝒇(𝒙)
Thus, 𝒇(𝒙) = 𝟑𝒙⁴ − 𝟐𝒙² + 𝟕 is even
(b)

𝟑 −𝒙 𝟑𝒙
𝒇 −𝒙 = 2
= − 2 = −𝒇(𝒙)
−𝒙 + 𝟏 𝒙 + 𝟏
𝟑𝒙
Thus, 𝒇 𝒙 = 2 is an odd function
𝒙 +𝟏
(c)
𝒇(−𝒙) = sin(−𝒙) + cos(−𝒙) = −sin 𝒙 + cos 𝒙
Since 𝒇(−𝒙) is not equal to 𝒇(𝒙) or −𝒇(𝒙)
⟹ 𝒇(𝒙) = sin 𝒙 + cos 𝒙 is neither even nor odd
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