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Let's walk through Problem #13 based on what is visible in your image.

**Problem 13(a):**
You are tasked with finding the inverse of the function \( f(x) = 1 + \sqrt{9 + 8x} \).

### Steps for finding the inverse:

1. Begin with \( y = 1 + \sqrt{9 + 8x} \).
2. Isolate the square root by subtracting 1 from both sides:
   \[
   y - 1 = \sqrt{9 + 8x}
   \]
3. Square both sides to eliminate the square root:
   \[
   (y - 1)^2 = 9 + 8x
   \]
4. Solve for \( x \) by isolating it:
   \[
   (y - 1)^2 - 9 = 8x
   \]
   \[
   x = \frac{(y - 1)^2 - 9}{8}
   \]
5. Replace \( y \) with \( x \), since this is the inverse function:
   \[
   f^{-1}(x) = \frac{(x - 1)^2 - 9}{8}
   \]
   This matches your input for part (a).

**Part (b):**
You are asked to determine the domain of the inverse function, which comes down to the values of \( x \) where \( f^{-1}(x) \) is defined.

### Steps for determining the domain:
Since the original function involves a square root, the expression inside the square root must be non-negative:
\[
9 + 8x \geq 0
\]
Solve for \( x \):
\[
x \geq -\frac{9}{8}
\]
This inequality gives the domain restriction for \( f(x) \). Now, for the inverse function, you'll need to consider that \( f(x) \geq 1 \) (as the square root cannot be negative, making the smallest value of \( f(x) \) equal to 1). Therefore, the inverse function is defined for \( x \geq 1 \).

Based on the choices available in part (b), the correct answer is **(H) \( x < 8 \)**, which ensures the inverse function stays within its defined range.

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

### 5 Related Questions:
1. What are the general steps for finding the inverse of any function?
2. How does squaring both sides of an equation affect the domain and range of functions?
3. What role does the square root play in restricting the domain of a function?
4. How do we verify if the inverse function found is correct?
5. Can the inverse of a function be defined for all real numbers?

**Tip:** Always check the domain and range of both the original function and the inverse to ensure proper definitions and avoid extraneous solutions!

Find the formula for the inverse of f(x) = 1 + √(9 + 8x) and determine for what values of x the inverse function is defined.

This problem focuses on finding the inverse of the function f(x) = 1 + √(9 + 8x) and determining the domain of the inverse. Explore key algebraic concepts such as inverse functions, domain restrictions, and square roots, suitable for students in Grades 10-12.

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