https://file.aiquickdraw.com/mathbot/file/1726846976596pplfmk3x.jpg

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We are given the functions:
- \( p(x) = 8x - 1 \)
- \( q(x) = \frac{x + 1}{8} \)

The goal is to determine if \( p(x) \) and \( q(x) \) are inverse functions by computing their composition, i.e., \( p(q(x)) \) and \( q(p(x)) \), and checking if both yield \( x \), indicating that they are inverses.

### Step 1: Compute \( p(q(x)) \)

We substitute \( q(x) \) into \( p(x) \):
\[
p(q(x)) = 8 \left( \frac{x + 1}{8} \right) - 1
\]
Simplifying:
\[
p(q(x)) = (x + 1) - 1 = x
\]

### Step 2: Compute \( q(p(x)) \)

Now, substitute \( p(x) \) into \( q(x) \):
\[
q(p(x)) = \frac{(8x - 1) + 1}{8}
\]
Simplifying:
\[
q(p(x)) = \frac{8x}{8} = x
\]

Since both compositions \( p(q(x)) \) and \( q(p(x)) \) yield \( x \), we can conclude that \( p(x) \) and \( q(x) \) are indeed inverse functions.

Thus, the correct choices to complete the statement are:
- The composition of \( p \) and \( q \), \( p \circ q \), **equals** \( x \), indicating that \( p \) and \( q \) **are** inverse functions.

Would you like any further details or clarification? 

Here are 5 related questions:
1. What are the properties of inverse functions?
2. How can you graphically interpret inverse functions?
3. What are the conditions for two functions to be inverses?
4. How does the composition of two functions help determine if they are inverses?
5. Can every function have an inverse?

**Tip**: Inverse functions reverse the effect of each other, and their graphs are reflections of each other over the line \( y = x \).

Let p(x) = 8x - 1 and q(x) = (x + 1)/8. Select the correct choices to complete the statement: The composition of p and q, p ∘ q (Select) in the function f(x) = x, indicating that p and q (Select) inverse functions.

Learn how to verify that two functions, p(x) = 8x - 1 and q(x) = (x + 1)/8, are inverse functions by checking their compositions. This algebra problem covers function composition and the inverse function theorem, suitable for students in Grades 9-12.

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