To satisfy the function composition f(g(x)) = √((x - 6)/(x - 7)), let f(x) = x^(1/6) and g(x) = (x - 6)/(x - 7). These functions, when composed, yield the sixth root of the fraction as required.
Step-by-step explanation:
The student is asking to find two nontrivial functions f(x) and g(x) such that when f(x) is composed with g(x) (written f(g(x))), the result is equal to the sixth root of the fraction ((x - 6)/(x - 7)). This is a problem of function composition and finding functions that meet certain criteria. To solve it, let's assume f(x) is the sixth root function, so f(x) = (x)^(1/6). Then g(x) would need to produce the inside of the sixth root, so g(x) = (x - 6)/(x - 7). When this g(x) is plugged into f(x), the result is √((x - 6)/(x - 7)) or (x - 6)/(x - 7))^(1/6) as required.