Final answer:
To satisfy the equation f(g(x)) = 8/(x-6)⁴ * f(x), we can define f(x) as x and g(x) as 8/(x-6)⁴ with a condition that x is not equal to 6 to avoid division by zero. By substituting g(x) into f(x), we fulfill the original equation, showing that these functions are a valid solution.
Step-by-step explanation:
To find two nontrivial functions f(x) and g(x) so that f(g(x)) = 8/(x-6)⁴ f(x), we must construct f(x) and g(x) to satisfy this functional equation. We need to consider the composite function and its result, which in this case is 8/(x-6)⁴ multiplied by f(x).
Let us choose g(x) = 8/(x-6)⁴. This is a simple function that will output the desired form when applied to x. Next, we need f(x) such that, when g(x) is inputted into f(x), it yields the multiplication of the original g(x) by f(x). A function that simply returns its input would satisfy this, but to fulfill the requirement of being nontrivial, let's define f(x) = x/c where c is a nonzero constant. Substituting g(x) into f(x) gives us f(g(x)) = g(x)/c.
After choosing a value for c, such as 1, we can verify: f(g(x)) = (8/(x-6)⁴)/c. To fulfill the original equation, c must be 1, so f(g(x)) = 8/(x-6)⁴ * f(x). Hence, with c = 1, f(x) = x and g(x) = 8/(x-6)⁴, the equation is satisfied.