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.