COMPOSITION OF FUNCTIONS EXAMPLES

Example 1 :

If f(x) = 2x2 + 3 and g(x) = x + 2, find f o g.

Solution :

f o g = f[g(x)]

= f[x + 2]

= 2(x + 2)2 + 3

= 2(x2 + 4x + 4) + 3

= 2x2 + 8x + 8 + 3

= 2x2 + 8x + 11

Example 2 :

If f(x) = 5x and g(x) = x + 2, find f o g(x2).

Solution :

f o g(x2) = f[g(x2)]

= f[x+ 2]

= 5(x2 + 2)

= 5x2 + 10

Example 3 :

If f(x) = 5x + 3 and g(x) = 7x - 2, find f o g(3).

Solution :

 f o g(x) = f[g(x)]

= f[7x - 2]

= 5(7x - 2) + 3

= 35x - 10 + 3

f o g(x) = 35x - 7

f o g(3) = 35(3) - 7

= 105 - 7

= 98

Example 4 :

Using f(x) = 4x + 3 and g(x) = x - 2, find f[g(5)].

Solution :

f[g(x)] = f[x - 2]

= 4(x - 2) + 3

= 4x - 8 + 3

    f[g(x)] = 4x - 5

f[g(5) = 4(5) - 5

= 20 - 5

= 15

Example 5 :

Using f(x) = 6x2 and g(x) = 14x + 4, find g o f.

Solution :

g o f = g[f(x)]

= g[6x2]

= 84x2 + 4

Example 6 :

Using f(x) = 5x + 4 and g(x) = x - 3, find g o f(6).

Solution :

g o f(x) = g[f(x)]

= g[5x + 4]

= (5x + 4) - 3

= 5x + 4 - 3

g o f(x) = 5x + 1

g o f(6) = 5(6) + 1

= 30 + 1

= 31

Example 7 :

If f(x) = x - 5 and g(x) = 2x + 3, verify f o g = g o f.

Solution :

f o g = [g(x)]

= f[2x + 3]

= (2x + 3) - 5

= 2x + 3 - 5

f o g = 2x - 2 ---->(1)

g o f = g[f(x)]

= g[x - 5]

= 2(x - 5) + 3

= 2x - 10 + 3

= 2x - 7 ---->(2)

From (1) and (2), we see that f o g ≠ g o f.

Example 8 :

Let f(x) = x + k and g(x) = 7x. If  f o g(2) = 7, find k. 

Solution :

f o g(2) = 7

f[g(2)] = 7

f[7(2)] = 7

f(14) = 7

14 + k = 7

k = -7

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