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determine whether each pair of functions are inverse functions. write yes or no

To determine whether the functions \( f(x) = 8x - 10 \) and \( g(x) = \frac{1}{8}x + \frac{5}{4} \) are inverse functions, we need to check if \( f(g(x)) = x \) and \( g(f(x)) = x \).

### Step 1: Compute \( f(g(x)) \)
First, substitute \( g(x) \) into \( f(x) \):
\[
f(g(x)) = 8 \left(\frac{1}{8}x + \frac{5}{4}\right) - 10
\]
Simplify the expression:
\[
= 8 \times \frac{1}{8}x + 8 \times \frac{5}{4} - 10
\]
\[
= x + 10 - 10
\]
\[
= x
\]

### Step 2: Compute \( g(f(x)) \)
Next, substitute \( f(x) \) into \( g(x) \):
\[
g(f(x)) = \frac{1}{8} \times (8x - 10) + \frac{5}{4}
\]
Simplify the expression:
\[
= \frac{1}{8} \times 8x - \frac{1}{8} \times 10 + \frac{5}{4}
\]
\[
= x - \frac{10}{8} + \frac{5}{4}
\]
Since \( \frac{10}{8} = \frac{5}{4} \):
\[
= x - \frac{5}{4} + \frac{5}{4}
\]
\[
= x
\]

### Conclusion:
Since both \( f(g(x)) = x \) and \( g(f(x)) = x \), the functions \( f(x) \) and \( g(x) \) are indeed inverse functions.

**Answer: Yes**

Would you like more details or have any questions?

Here are five related questions you might find interesting:

1. How do you find the inverse of a linear function?
2. What is the geometric interpretation of inverse functions on a graph?
3. How do you verify if two functions are inverses using their compositions?
4. What are the conditions under which a function has an inverse?
5. How can you determine if a function is one-to-one?

**Tip:** When checking if two functions are inverses, always verify both \( f(g(x)) = x \) and \( g(f(x)) = x \). If either fails, they are not inverses.

Determine whether each pair of functions are inverse functions. Write yes or no.

Learn how to determine if two functions are inverse functions using function composition. This problem involves algebraic concepts such as inverse functions and function composition, suitable for students in Grades 9-12.

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