https://file.aiquickdraw.com/mathbot/file/17249338657101oh5gwn3.jpg

To solve the given problem, let's break it down step by step.

The problem defines two functions:

\[
f(x) = 5x
\]
\[
g(x) = 3x - 2
\]

You are asked to find:

1. \( f(g(x)) \)
2. \( g(f(x)) \)
3. \( f(g(-1)) \)

### 1. Finding \( f(g(x)) \):
To find \( f(g(x)) \), substitute \( g(x) = 3x - 2 \) into the function \( f(x) = 5x \).

\[
f(g(x)) = f(3x - 2) = 5(3x - 2) = 15x - 10
\]

### 2. Finding \( g(f(x)) \):
To find \( g(f(x)) \), substitute \( f(x) = 5x \) into the function \( g(x) = 3x - 2 \).

\[
g(f(x)) = g(5x) = 3(5x) - 2 = 15x - 2
\]

### 3. Finding \( f(g(-1)) \):
First, evaluate \( g(-1) \):

\[
g(-1) = 3(-1) - 2 = -3 - 2 = -5
\]

Now, substitute \( -5 \) into \( f(x) = 5x \):

\[
f(g(-1)) = f(-5) = 5(-5) = -25
\]

### Summary of Results:
- \( f(g(x)) = 15x - 10 \)
- \( g(f(x)) = 15x - 2 \)
- \( f(g(-1)) = -25 \)

Let me know if you would like further explanation or have any questions. Here are some related questions for further practice:

1. What is \( g(f(-1)) \)?
2. Calculate \( f(g(0)) \).
3. What is the derivative of \( f(g(x)) \) with respect to \( x \)?
4. Solve \( f(g(x)) = 0 \) for \( x \).
5. If \( h(x) = f(x) + g(x) \), what is \( h(g(x)) \)?

**Tip:** When working with composite functions, remember to carefully substitute the inner function into the outer function and simplify before solving.

Learn how to solve composite functions problems involving f(g(x)) and g(f(x)) step-by-step. This problem explores the concepts of functions and composite functions using specific formulas f(x) = 5x and g(x) = 3x - 2. Suitable for high school students.

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