The problem presented in the image involves two functions, $f(x) = 2x + 5$ and $g(x) = -3x - 1$, and asks for $g(f(x))^{-1}$.

Let's break it down and solve it step by step.

Step 1: Compute $g(f(x))$

First, we substitute $f(x)$ into $g(x)$. This means replacing $x$ in $g(x)$ with $f(x)$.

Given:

• $f(x) = 2x + 5$
• $g(x) = -3x - 1$

We substitute $f(x)$ into $g(x)$: $g(f(x)) = -3(2x + 5) - 1$ Distribute the $-3$ across the terms inside the parentheses: $g(f(x)) = -6x - 15 - 1$ $g(f(x)) = -6x - 16$

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

To find $g(f(x))^{-1}$, we are looking for the inverse function. Let $y = g(f(x)) = -6x - 16$, and solve for $x$ in terms of $y$.

Starting with: $y = -6x - 16$ Solve for $x$: $y + 16 = -6x$ $x = \frac{-(y + 16)}{6}$ Thus, the inverse function is: $g(f(x))^{-1} = \frac{-(y + 16)}{6}$

Let me know if you'd like further details on this solution or need additional explanations.

Here are five related questions:

1. How do we generally compute the composition of functions $g(f(x))$?
2. What are the key steps to finding the inverse of a linear function?
3. Can the inverse process be applied to quadratic or other non-linear functions?
4. How does the chain rule apply when differentiating a composition of functions like $g(f(x))$?
5. What is the domain and range of $g(f(x))$ and its inverse?

Tip: When finding the inverse of a function, always ensure that the function is one-to-one (injective), as only such functions have inverses.