# HOW TO FIND THE RANGE OF COMPOSITE FUNCTIONS

The set of all images of the elements of X under f is called the ‘range’ of f.

The range of a function is a subset of its co-domain.

Question 1 :

Let A, B, C  N and a function f : A -> B be defined by f(x) = 2x + 1 and g : B -> C be defined by g(x) = x2 . Find the range of f o g and g o f .

Solution :

f o g  =  f[g(x)]

=  f[x2]

Now we apply x2 instead of x in f(x).

f o g  =  2 x2 + 1

y = 2x2 + 1

Range :

{y | y = 2x2 + 1 and x ∊ N}

g o f  =  g[f(x)]

=  g[2x + 1]

Now we apply 2x + 1 instead of x in g(x).

g o f  =  (2x + 1)2

Range :

{y | y = (2x + 1)2 and x ∊ N}

Question 2 :

Let f (x) = x2 −1 . Find (i) f o f (ii) f o f o f

Solution :

(i) f o f

=  f[f(x)]

=  f[x2 −1]

Now we apply x2 −1 instead of x in f(x).

=  (x2 −1)2 - 1

=  x4 - 2x2 + 1 - 1

f o f  =  x4 - 2x2

(ii) f o f o f

f o f  =  x4 - 2x2

f o f o f  =  f [f o f]

=  f[x4 - 2x2]

Now we apply x4 - 2x2 instead of x in f(x).

=  (x4 - 2x2)2 - 1

Question 3 :

If f : R -> R and g : R -> R are defined by f(x) = x5 and g(x) = x4 then check if f, g are one-one and f o g is one-one?

Solution :

f(x) = x5

For every positive and negative values of x, we get positive and negative values of y.

Every element in x is associated with different elements of y. Hence it is one to one function.

g(x) = x4

For every positive and negative values of x, we get only positive values of y.

Negative values of y is not associated with any elements of x. Hence it is not one to one function.

fog(x)  =  f[g(x)]

=  f[x4]

now, we apply x4 instead of x in f(x)

f[x4]  =  (x5)4

fog(x)  =  x20

fog is not one to one function.

Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here.

Kindly mail your feedback to v4formath@gmail.com

## Recent Articles

1. ### First Fundamental Theorem of Calculus - Part 1

Apr 17, 24 11:27 PM

First Fundamental Theorem of Calculus - Part 1

2. ### Polar Form of a Complex Number

Apr 16, 24 09:28 AM

Polar Form of a Complex Number