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Mathematics (Maths) Class 11

JEE : Doc: Some functions and their Graphs JEE Notes | EduRev

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Different types of functions and their graphical representation.

Identity function

A function f:R → R is said to be an identity function if f(x) = x, ∀ x ϵ R denoted by IR.

Let A = {1, 2, 3}

The function f: A → A defined by f(x) = x is an identity function.

f = {(1,1), (2,2), (3,3)}.

Doc: Some functions and their Graphs JEE Notes | EduRev


The graph of an identity function is a straight line passing through the origin.Doc: Some functions and their Graphs JEE Notes | EduRev

Each point on this line is equidistant from the coordinate axes.

The straight line makes an angle of 45° with the coordinate axes.

Linear function
A function f:R → R is said to be a linear function if f (x) = ax + b, where a ≠ 0, a and b are real constants, and x is a real variable.

Consider the linear function, f(x) = 3x + 7, ∀ x ϵ R

The ordered pairs satisfying the linear function are (0, 7), (-1, 4), (-2, 1).

On plotting these points on the Cartesian plane and then joining them, we get the graph of the linear function f of x is equal to 3x + 7, ∀ x ϵ R as shown in the figure.

Doc: Some functions and their Graphs JEE Notes | EduRev

Constant function
A function f:R → R is said to be a constant function, if f(x) = c , ∀ x ϵ R, where c is a constant.

Let f:R → R be a constant function defined by f(x) = 4 , ∀ x ϵ R.

The ordered pairs satisfying the linear function are: (0, 4), (-1, 4), (2, 4).

If the range of a function is a singleton set, then it is known as a constant function.

On plotting these points on the Cartesian plane and then joining them, we get the graph of the constant function f of x = 4,∀ x ϵ R as shown.

Doc: Some functions and their Graphs JEE Notes | EduRev

Also, from the graph, we can conclude that the graph of a constant function, f(x) = c, is always a straight line parallel to the X-axis, intersecting the Y-axis at (0, c).

Polynomial function
A function f:R → R is said to be a polynomial function if for all x in R, y = f(x) = a0+ a1x + a2x2 + ……………..+ anxn, where n is a non-negative integer and a0, a1, a2…….. an are real numbers.

Examples of polynomial functions

f(x) = x2 + 5x + 6 ,∀ x ϵ R

and f (x) = x3 + 4x + 2 ,∀ x ϵ R

Note: In a polynomial function, the powers of the variables should be non-negative integers.For example, f(x) = √x + 2 (∀ x ϵ R) is not a polynomial function because the power of x is a rational number.

Consider the polynomial function, f(x) = 3x2 +2x -3 ,∀ x ϵ RThe ordered pairs satisfying the polynomial function are (0, -3), (-1, -2), (1, 2), (2, 13), (-2, 5).

On plotting these points on the Cartesian plane and then joining them, we get the graph of the polynomial function f of x is equal to 3 x2 + 2 x - 3 ,∀ x ϵ R as shown.

Doc: Some functions and their Graphs JEE Notes | EduRev


Rational function
If f(x) and g(x) be two polynomial functions, then f(x)/g(x) such that g(x) ≠ 0 and ∀ x ϵ R, is known as a rational function.

Let us consider the function f ( x) = 2x -5/3x - 2 (x ≠ ⅔).

The ordered pairs satisfying the polynomial function are: (0, 5), (2, -¼), (1, -3).

On plotting these points on the Cartesian plane and then joining them, we get the graph of the given rational function as shown.
Doc: Some functions and their Graphs JEE Notes | EduRev


Modulus function
A function f: R → R defined by f(x) = |x| (∀ x ϵ R) is known as a modulus function.

If x is negative, then the value of the function is minus x, and if x is non-negative, then the value of the function is x. i.e. f(x) = x if x ≥ 0 = - x if x < 0.

The ordered pairs satisfying the polynomial function are (0, 0), (-1, 1), (1, 1), (-3, 3), (3, 3).

On plotting these points on the Cartesian plane and then joining them, we get the graph of modulus function f of x is equal to mod of x.
Doc: Some functions and their Graphs JEE Notes | EduRev

Greatest integer function
A function f: R → R defined by f(x) = [x],∀ x ϵ R assumes the value of the greatest integer, less than or equal to x.

From the definition of [x], we can see that

[x] = -1 for -1 £ x < 0

[x] = 0 for 0 £ x < 1

[x] = 1 for 1 £ x < 2

[x] = 2 for 2 £ x < 3, and so on.

⇒  f(2.5) will give the value 2 and f(1.2) will give the value 1, and so on…

Hence, the graph of the greatest integer function is as shown.
Doc: Some functions and their Graphs JEE Notes | EduRev

Signum Function

A function f: R→ R defined by

f(x) = { 1, if x > 0; 0, if x = 0; -1, if x < 0

Signum or the sign function extracts the sign of the real number and is also known as step function.

Doc: Some functions and their Graphs JEE Notes | EduRev



Algebra of Real Functions

Real-valued Mathematical Functions

In mathematics, a real-valued function is a function whose values are real numbers. It is a function that maps a real number to each member of its domain. Also, we can say that a real-valued function is a function whose outputs are real numbers i.e., f: R→R (R stands for Real).

Doc: Some functions and their Graphs JEE Notes | EduRev
Algebra of Real Functions
In this section, we will get to know about addition, subtraction, multiplication, and division of real mathematical functions with another.


Addition of Two Real Functions
Let f and g be two real valued functions such that f: X→R and g: X→R where X ⊂ R. The addition of these two functions (f + g): X→R  is defined by:

(f + g) (x) = f(x) + g(x), for all x ∈ X.

Subtraction of One Real Function from the Other
Let f: X→R and g: X→R be two real functions where X ⊂ R. The subtraction of these two functions (f – g): X→R  is defined by:

(f – g) (x) = f(x) – g(x), for all x ∈ X.

Multiplication by a Scalar

Let f: X→R be a real-valued function and γ be any scalar (real number). Then the product of a real function by a scalar γf: X→R is given by:

(γf) (x) = γ f(x), for all x ∈ X.

Multiplication of Two Real Functions

The product of two real functions say, f and g such that f: X→R and g: X→R, is given by

(fg) (x) = f(x) g(x), for all x ∈ X.

Division of Two Real Functions

Let f and g be two real-valued functions such that f: X→R and g: X→R where X ⊂ R. The quotient of these two functions (f  ⁄ g): X→R  is defined by:

(f / g) (x) = f(x) / g(x), for all x ∈ X.

Note: It is also called point wise multiplication.


Solved Example for You

Problem: Let f(x) = x3 and g(x) = 3x + 1 and a scalar, γ= 6. Find

1. (f + g) (x)

2. (f – g) (x)

3. (γf) (x)

4. (γg) (x)

5. (fg) (x)

6. (f / g) (x)

Solution: 
We have,

1. (f + g) (x) = f(x) + g(x) = x+ 3x + 1.

2. (f – g) (x) = f(x) – g(x) = x– (3x + 1) = x– 3x – 1.

3. (γf) (x) = γ f(x) = 6x

4. (γg) (x) = γ g(x) = 6 (3x + 1) = 18x + 6.

5. (fg) (x) = f(x) g(x) = x(3x +1) = 3x4 + x3.

6. (f / g) (x) = f(x) / g(x) = x/ (3x +1), provided x ≠ – 1/3.

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