https://file.aiquickdraw.com/mathbot/file/17264939008526wudprcr.jpg

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.

Given f(x) = 2x + 5 and g(x) = -3x - 1, find g(f(x))^{-1}.

Learn how to solve the composition of functions g(f(x)) where f(x) = 2x + 5 and g(x) = -3x - 1, and find its inverse. A step-by-step guide to function composition and calculating the inverse of a linear function, suitable for high school students.

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