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The question asks to find the inverse function \( f^{-1}(-2) \), where the function \( f(x) \) is given as:

\[
f(x) = \frac{x + 2}{x + 10}
\]

### Step 1: Find the inverse function

To find the inverse function, follow these steps:

1. Start by setting \( y = f(x) \):
   \[
   y = \frac{x + 2}{x + 10}
   \]

2. Swap \( y \) and \( x \) to begin solving for \( y \):
   \[
   x = \frac{y + 2}{y + 10}
   \]

3. Solve for \( y \):
   Multiply both sides by \( y + 10 \) to eliminate the denominator:
   \[
   x(y + 10) = y + 2
   \]
   Expand the left-hand side:
   \[
   xy + 10x = y + 2
   \]
   Get all terms involving \( y \) on one side:
   \[
   xy - y = 2 - 10x
   \]
   Factor out \( y \) on the left-hand side:
   \[
   y(x - 1) = 2 - 10x
   \]
   Solve for \( y \):
   \[
   y = \frac{2 - 10x}{x - 1}
   \]

Thus, the inverse function is:

\[
f^{-1}(x) = \frac{2 - 10x}{x - 1}
\]

### Step 2: Find \( f^{-1}(-2) \)

Now, substitute \( -2 \) into the inverse function:

\[
f^{-1}(-2) = \frac{2 - 10(-2)}{-2 - 1}
\]

Simplify:

\[
f^{-1}(-2) = \frac{2 + 20}{-3} = \frac{22}{-3} = -\frac{22}{3}
\]

### Final Answer:
\[
f^{-1}(-2) = -\frac{22}{3}
\]

Let me know if you need more details or if you have any questions!

Here are five related questions for further understanding:
1. How do you verify if two functions are inverses of each other?
2. What conditions must a function meet to have an inverse?
3. How do you graph the inverse of a function?
4. Can every rational function have an inverse? Why or why not?
5. What are the key differences between one-to-one and many-to-one functions?

**Tip:** Always check if the function is one-to-one before attempting to find the inverse. A function must pass the horizontal line test to ensure it has an inverse.

Find the inverse function value f^{-1}(-2) where f(x) = (x + 2) / (x + 10).

Step-by-step guide on how to find the inverse of the rational function f(x) = (x + 2) / (x + 10) and calculate f^{-1}(-2). Learn key concepts about inverse functions, solving rational equations, and the Inverse Function Theorem. Suitable for students in Grades 10-12.

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