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.