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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 \( x \).

Which of the following pairs of functions are inverses of each other?

Learn how to determine which pairs of functions are inverses of each other using composition of functions. This guide covers core concepts, including how to verify inverses through function composition, making it ideal for students in Grades 10-12.

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