Final answer:
A function has an inverse if it is one-to-one, which the given function f(x) is since it's an odd-degree polynomial. The derivative of f is f'(x) = (5/8)x^4 + (9/8)x^2, and the formula (f^(-1))'(f(x)) = 1/f'(x) is used to find the derivative of the inverse at a specific value.
Step-by-step explanation:
The student has asked to verify if the function f(x) = (1/8)(x^5 + 3x^3) has an inverse, and then to find the derivative of the inverse function, denoted as (f^(-1))'(a), at a specific value of a, which is -7. To determine if a function has an inverse, we must check if the function is one-to-one, meaning that each output is produced by only one input. In the case of f(x), it is a polynomial function of odd degree, which means it is indeed one-to-one and thus invertible.
Once we establish that f has an inverse, we can use the formula derived from the chain rule for derivatives that says (f^(-1))'(f(x)) = 1/f'(x). We need to calculate the derivative of f, which is f'(x) = (5/8)x^4 + (9/8)x^2, and then evaluate it at the x value that corresponds to f(x) = -7. After finding this x value, we can apply the formula to find the value of (f^(-1))'(-7).