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.