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."

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.