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The problem visible in the image is as follows:

**Problem #13:**
(a) Find a formula for the inverse of \( f(x) = 1 + \sqrt{9 + 8x} \). (Do not include \( f^{-1}(x) = \) in your answer.)

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

Let's go step by step to solve this.

### (a) Finding the inverse of \( f(x) = 1 + \sqrt{9 + 8x} \)
To find the inverse, we will:

1. Set \( y = f(x) \), so:
   \[
   y = 1 + \sqrt{9 + 8x}
   \]
2. Isolate the square root term 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:
   \[
   x = \frac{(y - 1)^2 - 9}{8}
   \]

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

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

The function \( f(x) = 1 + \sqrt{9 + 8x} \) involves a square root. For the square root to be defined, the expression inside must be non-negative:
\[
9 + 8x \geq 0
\]
Solving for \( x \):
\[
x \geq -\frac{9}{8}
\]

Therefore, \( f^{-1}(x) \) is defined for \( x \geq -\frac{9}{8} \).

Would you like more details on any part of the process or have any other questions?

### Related Questions:
1. How do you verify if a function and its inverse are correct?
2. What steps can you take to check if an inverse function is one-to-one?
3. Why is it important to ensure the square root expression is non-negative in functions like this?
4. How would you graph \( f(x) \) and its inverse on the same plane?
5. How does the domain of \( f(x) \) affect the range of its inverse?

**Tip:** When solving for the inverse of a function, always make sure to check if the inverse function satisfies the domain constraints of the original function.

Find a formula for the inverse of f(x) = 1 + sqrt(9 + 8x). (Do not include f^(-1)(x) = in your answer.) For what values of x is f^(-1)(x) defined?

Find the inverse of f(x) = 1 + sqrt(9 + 8x) and determine the domain where it is defined. This problem involves key algebraic concepts, including inverse functions and square root definitions. Suitable for Grades 11-12.

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