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Let's work through this problem step by step.

### Problem Overview:
You are given pairs of functions \( f(x) \) and \( g(x) \), and you need to:

1. Find \( f(g(x)) \) and \( g(f(x)) \) for each pair.
2. Determine if \( f \) and \( g \) are inverses of each other by checking if \( f(g(x)) = x \) and \( g(f(x)) = x \).

Let's address each part.

### Part (a):
#### Functions:
- \( f(x) = \frac{-4}{x} + 0 \), which simplifies to \( f(x) = \frac{-4}{x} \)
- \( g(x) = \frac{-4}{x} \)

#### Find \( f(g(x)) \):
Substitute \( g(x) \) into \( f(x) \):
\[
f(g(x)) = f\left(\frac{-4}{x}\right) = \frac{-4}{\frac{-4}{x}} = x
\]

#### Find \( g(f(x)) \):
Substitute \( f(x) \) into \( g(x) \):
\[
g(f(x)) = g\left(\frac{-4}{x}\right) = \frac{-4}{\frac{-4}{x}} = x
\]

Since both \( f(g(x)) = x \) and \( g(f(x)) = x \), the functions are inverses of each other. Therefore, for part (a), you select **"f and g are inverses of each other."**

### Part (b):
#### Functions:
- \( f(x) = 6x + 3 \)
- \( g(x) = \frac{x - 3}{6} \)

#### Find \( f(g(x)) \):
Substitute \( g(x) \) into \( f(x) \):
\[
f(g(x)) = f\left(\frac{x - 3}{6}\right) = 6\left(\frac{x - 3}{6}\right) + 3 = (x - 3) + 3 = x
\]

#### Find \( g(f(x)) \):
Substitute \( f(x) \) into \( g(x) \):
\[
g(f(x)) = g(6x + 3) = \frac{(6x + 3) - 3}{6} = \frac{6x}{6} = x
\]

Since both \( f(g(x)) = x \) and \( g(f(x)) = x \), the functions are inverses of each other. Therefore, for part (b), you select **"f and g are inverses of each other."**

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Would you like more details or have any specific questions?

### Related Questions:
1. What are the steps to verify if two functions are inverses?
2. Can two functions be inverses if only one composition equals \( x \)?
3. How does the domain affect the invertibility of functions?
4. How do transformations (like translations) affect inverse functions?
5. How do you find the inverse of a quadratic function?

### Tip:
For two functions to be inverses, \( f(g(x)) = x \) and \( g(f(x)) = x \) must both hold true for all values in their domains.

For each pair of functions f and g below, find f(g(x)) and g(f(x)). Then, determine whether f and g are inverses of each other.

Learn how to find the composition of two functions f(g(x)) and g(f(x)) and verify if they are inverses. This example includes functions such as f(x) = -4/x and g(x) = -4/x, suitable for high school algebra students.

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