The problem asks to find $(g^{-1} \circ f^{-1})(-3)$ using the functions:

• $f(x) = \frac{1}{8}x - 3$
• $g(x) = x^3$

We will solve this step by step.

Step 1: Find the inverse of $f(x)$

Given $f(x) = \frac{1}{8}x - 3$, we want to find $f^{-1}(x)$.

1. Set $y = \frac{1}{8}x - 3$.
2. Solve for $x$ in terms of $y$:

$y + 3 = \frac{1}{8}x$ $x = 8(y + 3)$

Thus, $f^{-1}(x) = 8(x + 3)$.

Step 2: Find the inverse of $g(x)$

For $g(x) = x^3$, we want to find $g^{-1}(x)$.

1. Set $y = x^3$.
2. Solve for $x$ in terms of $y$:

$x = \sqrt[3]{y}$

Thus, $g^{-1}(x) = \sqrt[3]{x}$.

Step 3: Compute $(g^{-1} \circ f^{-1})(-3)$

This means we first apply $f^{-1}$ to $-3$, and then apply $g^{-1}$ to the result.

1. Compute $f^{-1}(-3)$:

$f^{-1}(-3) = 8(-3 + 3) = 8(0) = 0$

2. Compute $g^{-1}(0)$:

$g^{-1}(0) = \sqrt[3]{0} = 0$

Thus, $(g^{-1} \circ f^{-1})(-3) = 0$.

The answer is $\boxed{0}$.

Would you like further explanation or have any questions?

Here are 5 related questions:

1. How do you find the inverse of a linear function?
2. What are the steps to find the inverse of a cubic function?
3. Why do we use function composition in mathematics?
4. What are the properties of inverse functions?
5. How do you verify if two functions are inverses of each other?

Tip: Always check that the domain and range of the function match the domain and range of its inverse when composing them.