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### Problem:
We are tasked with the following questions based on the function \( f(x) = 1 + \sqrt{9 + 8x} \):

1. **Part (a):** Find a formula for the inverse of \( f(x) \).
2. **Part (b):** Determine for which values of \( x \) the inverse function \( f^{-1}(x) \) is defined.

### Part (a): Finding the inverse of \( f(x) \)

The function is given as:
\[
f(x) = 1 + \sqrt{9 + 8x}
\]

To find the inverse, follow these steps:

1. Replace \( f(x) \) with \( y \):
   \[
   y = 1 + \sqrt{9 + 8x}
   \]

2. Solve for \( x \) in terms of \( y \):

   \[
   y - 1 = \sqrt{9 + 8x}
   \]

   Square both sides to remove the square root:
   \[
   (y - 1)^2 = 9 + 8x
   \]

3. Isolate \( x \):
   \[
   8x = (y - 1)^2 - 9
   \]
   \[
   x = \frac{(y - 1)^2 - 9}{8}
   \]

So, the inverse function \( f^{-1}(x) \) is:
\[
f^{-1}(x) = \frac{(x - 1)^2 - 9}{8}
\]

### Part (b): For what values of \( x \) is \( f^{-1}(x) \) defined?

The original function involves a square root, so we must ensure the expression inside the square root is non-negative. For the function \( f(x) = 1 + \sqrt{9 + 8x} \), the expression under the square root, \( 9 + 8x \), must be greater than or equal to zero.

Solve for \( x \):
\[
9 + 8x \geq 0
\]
\[
x \geq -\frac{9}{8}
\]

Therefore, the domain of the original function is \( x \geq -\frac{9}{8} \). Since the inverse function \( f^{-1}(x) \) must reflect this domain, the inverse function is defined for values of \( x \geq 1 \), because \( f(x) \geq 1 \).

### Summary of Answers:
- **(a)**: The inverse function is \( f^{-1}(x) = \frac{(x - 1)^2 - 9}{8} \).
- **(b)**: The inverse is defined for \( x \geq 1 \).

---

Would you like further clarification or additional details on any part? Here are some related questions:

1. How do you verify the correctness of an inverse function?
2. What is the graphical relationship between a function and its inverse?
3. Can you explain the domain and range restrictions for inverses?
4. How do inverse functions apply in real-world contexts?
5. What happens to the inverse when the original function is not one-to-one?

**Tip:** Remember that a function needs to be one-to-one (each \( y \) value is paired with one unique \( x \)) for its inverse to exist.

Find a formula for the inverse of f(x) = 1 + sqrt(9 + 8x). For what values of x is f^-1(x) defined?

Learn how to find the inverse of the function f(x) = 1 + sqrt(9 + 8x) with a detailed step-by-step solution. Understand the domain restrictions of the inverse function, f^-1(x). Suitable for high school students in Grades 10-12.

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