To determine which pair of functions are inverses of each other, we need to check if the composition of the two functions $f(g(x))$ and $g(f(x))$ equals $x$.

Let's evaluate each pair of functions:

Pair A:

$f(x) = \frac{7}{x} + 10$ and $g(x) = \frac{7}{x - 10}$

• $f(g(x)) = f\left(\frac{7}{x - 10}\right) = \frac{7}{\frac{7}{x - 10}} + 10 = x - 10 + 10 = x$
• $g(f(x)) = g\left(\frac{7}{x} + 10\right) = \frac{7}{\left(\frac{7}{x} + 10\right) - 10} = \frac{7}{\frac{7}{x}} = x$

So, $f(g(x)) = g(f(x)) = x$.

Pair B:

$f(x) = \frac{7}{x - 9}$ and $g(x) = \frac{7}{x} + 9$

• $f(g(x)) = f\left(\frac{7}{x} + 9\right) = \frac{7}{\left(\frac{7}{x} + 9\right) - 9} = \frac{7}{\frac{7}{x}} = x$
• $g(f(x)) = g\left(\frac{7}{x - 9}\right) = \frac{7}{\left(\frac{7}{x - 9}\right)} + 9 = x - 9 + 9 = x$

So, $f(g(x)) = g(f(x)) = x$.

Pair C:

$f(x) = 2x + 8$ and $g(x) = \frac{x - 8}{2}$

• $f(g(x)) = f\left(\frac{x - 8}{2}\right) = 2\left(\frac{x - 8}{2}\right) + 8 = x - 8 + 8 = x$
• $g(f(x)) = g\left(2x + 8\right) = \frac{2x + 8 - 8}{2} = x$

So, $f(g(x)) = g(f(x)) = x$.

Pair D:

$f(x) = 5x + 11$ and $g(x) = \frac{x - 11}{5}$

• $f(g(x)) = f\left(\frac{x - 11}{5}\right) = 5\left(\frac{x - 11}{5}\right) + 11 = x - 11 + 11 = x$
• $g(f(x)) = g\left(5x + 11\right) = \frac{5x + 11 - 11}{5} = x$

So, $f(g(x)) = g(f(x)) = x$.

Conclusion:

All the function pairs (A, B, C, and D) are inverse pairs since $f(g(x)) = g(f(x)) = x$ for each one.

Do you need any further details or have any questions about this? Here are some related questions:

1. How do you find the inverse of a function?
2. What is the relationship between the graph of a function and its inverse?
3. Can every function have an inverse? Why or why not?
4. How do domain and range change when taking the inverse of a function?
5. How can you verify algebraically that two functions are inverses?

Tip: To find the inverse of a function $f(x)$, swap $x$ and $y$, then solve for $y$ in terms of $x$.