Question

Given f(x)=x-4 and g(x)=x²-3x+ | algebraically determine all values of x such that f(g(x)) = g(f(x))

Answer 1

Explanation:

When we plug in the values we get

`f(g(x)) = ({x}^(2) - 3x + 1) - 4 \\ = {x}^(2) - 3x - 3`

`g(f(x)) = {(x - 4)}^(2) - 3(x - 4) + 1`

`{x}^(2) - 3x - 3 = {(x - 4)(x - 4)} - 3x + 12 + 1 \\ {x }^(2) - 3x - 3 = {x}^(2) - 8x + 16 - 3x + 13 \\ \\ {x}^(2) - {x}^(2) - 3x + 8x + 3x - 3 - 16 - 13 = 0 \\ 8x - 32 = 0 \\ 8x = 32 \\ (8x)/(8) = (32)/(8) \\ x = 4`