Composite Functions

A composite function is usually a function that is written inside another function. Let f(x) and g(x) be two functions, then gof(x) is a composite function. Let us discuss the definition of the basic composite function gof(x) and how f(x) and g(x) are related. The questions from this topic are frequently asked in JEE and other competitive examinations.

Let f: A→ B and g: B→ C be two functions, then gof: A→ C is defined by gof(x) = g(f(x)).

The domain of gof is the set of all numbers in the domain of f so that f(x) is in the domain of g.

The composite function gof(x) is read as “g of f of x”. If f(x)and g(x) are two functions, then fog(x), gof(x), gog(x) and fof(x) are composite functions.

• fog(x) = f(g(x))
• gof(x) = g(f(x))
• gog(x) = g(g(x))
• fof(x) = f(f(x))
• fogoh(x) = f(g(h(x)))
• fofof(x) = f(f(f(x)))

The order of the function is important in a composite function since (fog)(x) is not equal to (gof)(x).

How to Solve Composite Functions

Step 1: Write the composition fog(x) as f(g(x)).

Step 2: For every occurrence of x in the outside function, replace x with the inside function g(x).

Step 3: Simplify the function.

Consider the following example.

Let f(x) = 3x+4 and g(x) = x-2. Find fog(x).

Solution:

Given f(x) = 3x+4

g(x) = x-2

fog(x) = f(g(x))

Substitute g(x) in place of x in f(x).

=> f(x-2) = 3(x-2)+4

= 3x-6+4

= 3x-2

So f(g(x)) = 3x-2

Important Points to Remember

1) In general f(g(x)) ≠ g(f(x)).

2) Composite functions are associative. If h:A→ B, g: B→ C and f: C→ D, then

(fog)oh(x) = fo(goh)x.

3) If f(x) and g(x) are one-one functions, then gof(x) is also one-one function.

4) If f(x) and g(x) are on-to functions, then gof(x) is also on-to function.

5) If f(x) and g(x) are bijective functions, then gof(x) is also a bijective function.

Solved Examples

Question 1:

Let f(x) = x² and g(x) = √(1-x²). Find gof(x) and fog(x).

Solution:

Given f(x) = x²

g(x) = √(1-x²)

gof(x) = g(f(x))

= g(x²)

= √(1-(x²)²)

= √(1-x⁴)

gof(x) = √(1-x⁴)

fog(x) = f(g(x))

= f(√(1-x²))

= (√(1-x²))²

= 1-x²

So, fog(x) = 1-x².

Question 2:

For f(x) = 2x + 3 and g(x) = -x² + 1, then the composite function defined by (fog)(x) is

(a) -2x²+5

(b) 2x²-5

(c) 2x+1

(d) x²-1

Solution:

Given f(x) = 2x + 3

g(x) = -x² + 1

(f o g)(x) = f(g(x))

= f(-x²+1)

= 2(-x²+1) + 3

= -2x²+2+3

= -2x²+5

So, the answer is -2x²+5.

Hence, option a is the answer.

Question 3:

Given f(2) = 3, g(3) = 2, f(3) = 4 and g(2) = 5, find the value of fog(3).

(a) 3

(b) 7

(c) 4

(d) 12

Solution:

f(2) = 3

g(3) = 2

f(3) = 4

g(2) = 5

fog(3) = f(g(3))

= f(2) (since g(3) = 2)

= 3 (given)

So, fog(3) = 3.

Hence, option a is the answer.

Question 4:

Find g(f(5)) when f(x) = x + 3 and g(x) = x/2.

(a) 16

(b) 8

(c) 4

(d) 12

Solution:

Given f(x) = x + 3

g(x) = x/2

f(5) = 5+3 = 8

g(f(5)) = g(8)

= 8/2

= 4

Hence, option c is the answer.

Question 5:

If f(x) = sin^2⁡x and the composite function g(f(x)) = |sin⁡x|, then g(x) is equal to

(a) √(x-1)

(b) √x

(c) √(x+1)

(d) √-x

Solution:

Given f(x) = sin^2⁡x

g(f(x)) = |sin⁡x|

g(sin^2⁡x) = |sin⁡x|

= √sin^2⁡x

=> g(x) = √x

Hence, option b is the answer.

Video Lessons

Composite Functions

Composite and Periodic Functions