Notes On Functions and their Graphs - CBSE Class 11 Maths

A relation f from a set A to set B is said to be function if every element of set A has one and only one image in set B.

A function f is a relation if no two ordered pairs in the relation have the same first element.

f = {(1, 2), (2, 3), (3, 4), (4, 5), (5, 6), (6, 7)}

A function that is a relation between two sets can be represented graphically because it is the set of ordered pairs and can be plotted on the Cartesian plane.

The notation f : A
B means that f is a function from A to B.

A =Domain of f = {1, 2, 3, 4, 5, 6} and B = Co-domain of f = {2, 3, 4, 5, 6, 7}

Given a ϵ A, there is a unique element b in B, such that f (a) = b.

The set of all values of f (a) taken together is called the range of f or the image of A under f.

Symbolically, range of f = {b: b= f (a), for some a in A} = {2, 3, 4, 5, 6, 7}.

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)}.



The graph of an identity function is a straight line passing through the origin.



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.



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.



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 R

The 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.



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 ≠ ).

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.



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.


Signum function
A function f: RR defined by f(x) = x x is called a signum function.

It can also be defined as

1 if x > 0

f(x) = 0 if x = 0

-1 if x < 0 (,∀ x ϵ R)

The ordered pairs satisfying the signum function are (1, 1), (2, 1), (0, 0), (-1, -1), (-5, -1).

On plotting these points on the Cartesian plane and then joining them, we get the graph of signum function f of x is equal to mod of x divided by x.


The domain of the signum function is R, and the range is {-1,0,1}.

It is also written as sign (x).


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.

Summary

A relation f from a set A to set B is said to be function if every element of set A has one and only one image in set B.

A function f is a relation if no two ordered pairs in the relation have the same first element.

f = {(1, 2), (2, 3), (3, 4), (4, 5), (5, 6), (6, 7)}

A function that is a relation between two sets can be represented graphically because it is the set of ordered pairs and can be plotted on the Cartesian plane.

The notation f : A
B means that f is a function from A to B.

A =Domain of f = {1, 2, 3, 4, 5, 6} and B = Co-domain of f = {2, 3, 4, 5, 6, 7}

Given a ϵ A, there is a unique element b in B, such that f (a) = b.

The set of all values of f (a) taken together is called the range of f or the image of A under f.

Symbolically, range of f = {b: b= f (a), for some a in A} = {2, 3, 4, 5, 6, 7}.

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)}.



The graph of an identity function is a straight line passing through the origin.



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.



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.



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 R

The 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.



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 ≠ ).

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.



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.


Signum function
A function f: RR defined by f(x) = x x is called a signum function.

It can also be defined as

1 if x > 0

f(x) = 0 if x = 0

-1 if x < 0 (,∀ x ϵ R)

The ordered pairs satisfying the signum function are (1, 1), (2, 1), (0, 0), (-1, -1), (-5, -1).

On plotting these points on the Cartesian plane and then joining them, we get the graph of signum function f of x is equal to mod of x divided by x.


The domain of the signum function is R, and the range is {-1,0,1}.

It is also written as sign (x).


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.

Videos

References

Previous