COMPOSITION OF THREE FUNCTIONS

Consider the functions f(x), g(x) and h(x) as given below. Find (f o g) o h and f o (g o h) in each case and also show that (f o g) o h = f o (g o h).

Example 1 :

f(x) = x - 1 , g(x) = 3x + 1 and h(x) = x²

Solution :

f o (g o h) :

(f o g) o h :

From (1) and (2), 

f o (g o h) = (f o g) o h

Example 2 :

f(x) = x², g(x) = 2x and h(x) = x + 4

Solution :

f o (g o h) :

(f o g) o h :

From (1) and (2), 

f o (g o h) = (f o g) o h

Example 3 :

f(x) = x - 4, g(x) = x² and h(x) = 3x - 5

Solution :

f o (g o h) : 

(f o g) o h :

From (1) and (2), 

f o (g o h) = (f o g) o h

Example 4 :

Given f(x) = -8x², g(x) = -3x + 9 and h(x) = √x, find [(f + g)o h] (x)

Solution :

[(f + g)o h] (x) = (f + g)[h (x)]

= (f + g) (√x)

= f(√x) + g(√x)

f(x) + g(x) = -8x + (-3√x + 9)

= -8x - 3√x - 9

Example 5 :

Let f(x) = 3 - x² and h(x) = f(g(x)). Fill the table.

Solution :

h(x) = f(g(x))

When x = -1

h(-1) = f(g(-1))

Applying the value of g(-1), we get

= f(2)

Now using the given function f(x), let us evaluate f(2).

f(x) = 3 - x²

f(2) = 3 - 2²

= 3 - 4

= -1

Then, h(-1) = f(g(-1)) = -1

When x = 0

h(0) = f(g(0))

Applying the value of g(0), we get

= f(-1)

f(x) = 3 - (-1)²

f(-1) = 3 - 1

= 2

h(0) = f(g(0)) = 2

When x = 1

h(1) = f(g(1))

Applying the value of g(1), we get

= f(4)

f(x) = 3 - 4²

f(4) = 3 - 16

= -13

h(1) = f(g(1)) = -13

Example 6 :

Fill in the following table, given that h(x) = f(g(x))

Solution :

When x = -7

h(-7) = f(g(-7))

The value g(-7) is 6.

h(-7) = f(6)

h(-7) = 0

When x = -3

h(-3) = f(g(-3))

The value g(-3) is -7.

h(-3) = f(-7)

h(-3) = 2

When x = -1

h(-1) = f(g(-1))

The value g(-1) is -3.

h(-1) = f(-3)

h(-1) = 6

When x = 0

h(0) = f(g(0))

The value g(0) is 2.

h(0) = f(2)

h(0) = -7

When x = 2

h(2) = f(g(2))

The value g(2) is 5.

h(2) = f(5)

h(2) = -1

When x = 5

h(5) = f(g(5))

The value g(5) is 0.

h(5) = f(0)

h(5) = 5

When x = 6

h(6) = f(g(6))

The value g(6) is -1.

h(6) = f(-1)

h(6) = -3

Example 7 :

Given 𝑓(𝑥)  = 5𝑥 - 2𝑏 while 𝑔(𝑥) = 4𝑏𝑥. If 𝑓(𝑔(1)) = 36, what is 𝑔(𝑓(1)) ?

Solution :

Given that,

𝑓(𝑥)  = 5𝑥 - 2𝑏

𝑔(𝑥) = 4𝑏𝑥.

𝑓(𝑔(1)) = 36 -----(1)

g(1) = 4b(1) ==> 4b

Applying the value of g(1) in (1), we get

𝑓(4b) = 36

Applying x = 4b as x in the function f(x).

f(4b)  = 5(4b) - 2𝑏

= 20b - 2b

f(4b) = 18b

18b = 36

b = 36/18

b = 2

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