HOW TO FIND THE MISSING VALUE IN COMPOSITION OF TWO FUNCTIONS

Find the value of k, such that f o g = g o f.

Example 1 :

f(x) = 3x + 2 and g(x) = 6x - k

Solution :

f o g = g o f

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

f[6x - k] = g[3x + 2]

3(6x - k) + 2 = 6(3x + 2) - k

18x - 3k + 2 = 18x + 12 - k

Subtract 18x from each side. 

-3k + 2 = 12 - k

Add k to each side. 

-2k + 2 = 12

Subtract 2 from each side. 

-2k = 10

Divide each side by -2.

k = -5

Example 2 :

f(x) = 2x - k and g(x) = 4x + 5

Solution : 

f o g = g o f

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

f[4x + 5] = g[2x - k]

2(4x + 5) - k = 4(2x - k) + 5

8x + 10 - k = 8x - 4k + 5

Subtract 18x from each side. 

10 - k = -4k + 5

Add 4k to each side. 

10 + 3k = 5

Subtract 10 from each side. 

3k = -5

Divide each side by 3.

k = -5/3

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