Question:

Given:     f (x) = x + 4            g (x) = (x - 2)2          f (g (u (x))) = 4x2 - 8x + 8

Find       u (x) = ?

 

Solution:

Substitute u(x) or u in the composite function:  f (g (u (x)))

 f (g (u (x))) = (u-2)2 + 4 =  4x2 - 8x + 8

(u-2)2 =  4x2 - 8x + 8 - 4

(u-2)2 =  4x2 - 8x + 4

(u-2)2 =  4 (x2 - 2x + 1)

(u-2)2 =  4 (x - 1)2

There are two solutions:

Solution 1                            Solution 2              
u-2 =  + 2 (x - 1) 

u-2 =  2x - 2 

u =  2x

 u-2 =  - 2 (x - 1)

u-2 =  -2x + 2

u = - 2x + 4

                   

                   

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