Question

Given the function g(x) = 41x^3 + a for some constant a, which of the following describes the inverse function g^(-1)(x)?

A) No restriction on the domain of g(x); the inverse is g^(-1)(x) = 3.
B) No restriction on the domain of g(x); the inverse is g^(-1)(x) = a.
C) Domain of g(x) restricted to x = 0; the inverse is g^(-1)(x) = 7.
D) Domain of g(x) restricted to x ≥ 0; the inverse is g^(-1)(x) = 4.41.

Answer 1

The inverse function of
`g(x) = 41x^3 + a` is
`g^(^-^1^)(x) = ((x - a)/41)^(^1^/^3^)`.

Step-by-step explanation:

The given function is
`g(x) = 41x^3 + a`, where a is a constant. To find the inverse function, we need to swap the roles of x and y and solve for y. Let's denote the inverse function as
`g^(^-^1^)(x)`.

To find the inverse function, we start by replacing g(x) with y in the equation:

`x = 41y^3 + a`

Next, we solve for y and express it as a function of x:

`y = ((x - a)/41)^(^1^/^3^)`

Therefore, the inverse function is given by
`g^(^-^1^)(x) = ((x - a)/41)^(^1^/^3^)`