Find two nontrivial functions f ( x ) and g ( x ) so f ( g ( x ) ) = ( − 9 − 7 x ) ^6

Question

Find two nontrivial functions f ( x ) and g ( x ) so f ( g ( x ) ) = ( − 9 − 7 x ) ^6

f(x) =

g(x) =

Answer 1

Hi Nico,

There are many "correct" answers to this question, but the one that I think is most straightforward is:

f(x) = x^6

g(x) = -9 - 7x

To me, what makes this problem tricky/confusing is that you are required to reuse the variable name "x" in both "f(x)" and "g(x)". When you have a composite function f(g(x)), it is better to conceptualize this function as a two-step process: Given x, you first compute g, then using that g you compute f. So to me, a more clear notation would be

g(x) = -9 - 7x

f(g) = g^6.

For example, given x = 1, then g = -9 -7 * 1 = -16, and then f = (-16)^6 = 16,777,216.