To find the range and domain for the inverse of the function $f(x) = 8 - (x + 5)^2$, we need to follow a few steps.

Step 1: Determine the domain and range of $f(x)$

The given function $f(x) = 8 - (x + 5)^2$ is a downward-opening parabola. To find the domain and range of $f(x)$:

1. Domain of $f(x)$:
   The function $f(x) = 8 - (x + 5)^2$ is defined for all real numbers $x$. Therefore, the domain is: $\text{Domain of } f(x) = (-\infty, \infty)$

2. Range of $f(x)$:
   The vertex of the parabola $y = 8 - (x + 5)^2$ occurs at $x = -5$. At this point, $y = 8$. Since the parabola opens downwards, the maximum value is $8$ and it decreases without bound as $x$ moves away from $-5$. Therefore, the range is: $\text{Range of } f(x) = (-\infty, 8]$

Step 2: Find the inverse of $f(x)$

To find the inverse, we first set $y = f(x)$:

$y = 8 - (x + 5)^2$

Solving for $x$:

$y - 8 = -(x + 5)^2$

$-(y - 8) = (x + 5)^2$

$x + 5 = \pm \sqrt{8 - y}$

$x = -5 \pm \sqrt{8 - y}$

Since $f(x)$ is a function, its inverse must also be a function. To ensure $f^{-1}(x)$ is a function, we restrict the domain of $f(x)$ to where it is one-to-one. Considering $x \geq -5$:

$x = -5 + \sqrt{8 - y}$

Thus, the inverse function $f^{-1}(y)$ is:

$f^{-1}(y) = -5 + \sqrt{8 - y}$

Step 3: Determine the domain and range of $f^{-1}(y)$

1. Domain of $f^{-1}(y)$:
   The range of $f(x)$ becomes the domain of $f^{-1}(y)$. Therefore, the domain is: $\text{Domain of } f^{-1}(y) = (-\infty, 8]$

2. Range of $f^{-1}(y)$:
   The domain of $f(x)$ becomes the range of $f^{-1}(y)$. Therefore, the range is: $\text{Range of } f^{-1}(y) = [-5, \infty)$

Summary

• The domain of the inverse function $f^{-1}(y)$ is $(-\infty, 8]$.
• The range of the inverse function $f^{-1}(y)$ is $[-5, \infty)$.

Would you like more details or have any questions?

Here are some questions you might want to explore next:

1. How do you find the inverse of a different function?
2. What are the properties of inverse functions?
3. How do you verify that two functions are inverses of each other?
4. What are the transformations of the graph of $f(x)$?
5. How do the concepts of domain and range apply to real-world problems?

Tip: When finding the inverse of a function, always ensure that the original function is one-to-one on its domain, otherwise restrict the domain to make it one-to-one.