Function $g$ is defined as follows:
\[g(x) = \left\{
\begin{array}{cl}
5x^2 & \text{if } x \le -3, \\
21+x& \text{if }-3 < x \le 10, \\
2-\sqrt{x}& \text{if }x > 10.
\end{array}\right.\]This function has an inverse.

What is $g^{-1}(-7)$?

To find $ g^{-1}(-7) $, we need to determine the value of $ x $ for which $ g(x) = -7 $. We will check each piece of the piecewise function $ g(x) $ to find where this condition holds.

1. **Case 1: $ x \le -3 $**

   \[
   g(x) = 5x^2
   \]

   We set up the equation:

   \[
   5x^2 = -7
   \]

   Since $ 5x^2 \ge 0 $ for all real $ x $, there are no real solutions to this equation. Thus, $ -7 $ cannot be in this interval.

2. **Case 2: $ -3 < x \le 10 $**

   \[
   g(x) = 21 + x
   \]

   We set up the equation:

   \[
   21 + x = -7
   \]

   Solving for $ x $:

   \[
   x = -7 - 21
   \]

   \[
   x = -28
   \]

   However, $-28$ is not in the interval $ -3 < x \le 10 $. Thus, $ -7 $ cannot be in this interval either.

3. **Case 3: $ x > 10 $**

   \[
   g(x) = 2 - \sqrt{x}
   \]

   We set up the equation:

   \[
   2 - \sqrt{x} = -7
   \]

   Solving for $ \sqrt{x} $:

   \[
   2 + 7 = \sqrt{x}
   \]

   \[
   \sqrt{x} = 9
   \]

   Squaring both sides:

   \[
   x = 81
   \]

   Since $ 81 > 10 $, $ x = 81 $ is a valid solution in this interval.

Hence, the value of $ x $ for which $ g(x) = -7 $ is $ x = 81 $. Therefore, the inverse function value is:

\[
g^{-1}(-7) = 81
\]

Would you like more details or have any questions? Here are some related questions you might find interesting:

1. What is the domain of $ g(x) $?
2. What is the range of $ g(x) $?
3. Find $ g^{-1}(0) $.
4. Determine $ g^{-1}(15) $.
5. What are the properties of the inverse function $ g^{-1}(x) $?
6. How does the function $ g(x) $ change at the boundaries of the intervals?
7. Sketch the graph of $ g(x) $.
8. What is the value of $ g^{-1}(21) $?

**Tip:** When working with piecewise functions, always check each interval separately to ensure all possible cases are considered.

Learn how to find the inverse of the piecewise function g(x) defined by g(x) = 5x^2 for x <= -3, g(x) = 21 + x for -3 < x <= 10, and g(x) = 2 - sqrt(x) for x > 10. Explore step-by-step solutions and understand the intervals where each piece of the function applies. Suitable for advanced high school students studying piecewise functions and inverse functions.

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