Use the Inverse Function Property to determine whether *f* and *g* are inverses of each other.

*f*(*x*) = 

1/

*(x* − 10)

,   *x* ≠ 10;   *g*(*x*) = 

*1/x  + 10*
 

To determine whether \( f(x) \) and \( g(x) \) are inverses of each other, we can use the **Inverse Function Property**, which states:

If \( f \) and \( g \) are inverse functions, then:
\[
(f \circ g)(x) = f(g(x)) = x
\]
and
\[
(g \circ f)(x) = g(f(x)) = x
\]

### Step 1: Compute \( f(g(x)) \)
Given \( g(x) = \frac{1}{x} + 10 \), we substitute it into \( f(x) = \frac{1}{x - 10} \):

\[
f(g(x)) = f\left( \frac{1}{x} + 10 \right) = \frac{1}{\left( \frac{1}{x} + 10 \right) - 10} = \frac{1}{\frac{1}{x}} = x
\]

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

### Step 2: Compute \( g(f(x)) \)
Now, substitute \( f(x) = \frac{1}{x - 10} \) into \( g(x) = \frac{1}{x} + 10 \):

\[
g(f(x)) = g\left( \frac{1}{x - 10} \right) = \frac{1}{\frac{1}{x - 10}} + 10 = (x - 10) + 10 = x
\]

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

### Conclusion:
Since both \( f(g(x)) = x \) and \( g(f(x)) = x \), we can conclude that \( f(x) \) and \( g(x) \) are inverses of each other.

Would you like further details or have any questions?

Here are 5 related questions to expand your understanding:
1. What are the general conditions for two functions to be inverses of each other?
2. How can you visually verify if two functions are inverses using their graphs?
3. How does the domain and range of a function affect its inverse?
4. Can all functions have an inverse? Why or why not?
5. How does composition of functions help in verifying if two functions are inverses?

**Tip:** Inverse functions "undo" each other, meaning the output of one function becomes the input of the other, and vice versa.

Use the Inverse Function Property to determine whether f and g are inverses of each other. f(x) = 1/(x − 10), x ≠ 10; g(x) = 1/x + 10

This solution explains how to use the Inverse Function Property to determine if the functions f(x) = 1/(x - 10) and g(x) = 1/x + 10 are inverses. Learn the step-by-step process involving function composition and key mathematical concepts such as inverse functions and composition of functions. Suitable for Grades 10-12.

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