rewrite p(x)=$$ \left\vert x^2-36\right\vert $$-5 as three piecewise functions

To rewrite the function $ p(x) = \left\vert x^2 - 36 \right\vert - 5 $ as three piecewise functions, we need to consider the points where the expression inside the absolute value changes sign, specifically where $ x^2 - 36 = 0 $.

1. Solve $ x^2 - 36 = 0 $ to find the points of interest:
   \[
   x^2 - 36 = 0
   \]
   \[
   x^2 = 36
   \]
   \[
   x = \pm 6
   \]

These points divide the real number line into three intervals: $ (-\infty, -6) $, $ [-6, 6] $, and $ (6, \infty) $.

2. On each interval, determine the expression for $ p(x) $:

### Interval 1: $ x < -6 $
Here, $ x^2 - 36 > 0 $, so the absolute value can be removed without changing the sign:
\[
p(x) = (x^2 - 36) - 5 = x^2 - 41
\]

### Interval 2: $ -6 \leq x \leq 6 $
Here, $ x^2 - 36 \leq 0 $, so the absolute value changes the sign of the expression:
\[
p(x) = -(x^2 - 36) - 5 = -x^2 + 36 - 5 = -x^2 + 31
\]

### Interval 3: $ x > 6 $
Here, $ x^2 - 36 > 0 $, so the absolute value can be removed without changing the sign:
\[
p(x) = (x^2 - 36) - 5 = x^2 - 41
\]

Thus, the piecewise function is:

\[
p(x) = 
\begin{cases} 
x^2 - 41 & \text{if } x < -6 \\
-x^2 + 31 & \text{if } -6 \leq x \leq 6 \\
x^2 - 41 & \text{if } x > 6 
\end{cases}
\]

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

1. How does the absolute value function affect the behavior of $ p(x) $?
2. Can you graph the piecewise function to visualize it?
3. How do the different intervals affect the continuity of $ p(x) $?
4. What is the derivative of $ p(x) $ in each interval?
5. What happens if you change the constant $ -5 $ to another value?
6. How would you express $ p(x) $ if the function inside the absolute value were more complex?
7. Can $ p(x) $ have any local minima or maxima within the intervals?
8. How do you solve for points of intersection between $ p(x) $ and a linear function?

**Tip:** When dealing with absolute value functions, always break the function down into different cases based on where the expression inside the absolute value changes sign.

Learn how to rewrite the absolute value function p(x) = |x^2 - 36| - 5 as three piecewise functions based on the sign changes of the expression inside the absolute value. This problem covers key concepts in piecewise functions and absolute value functions. Suitable for advanced high school students.

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