https://file.aiquickdraw.com/mathbot/file/17284052794360xpkk2ve.jpg

The image shows piecewise functions labeled as problems 32, 33, and 34, which are defined by different expressions over specific intervals of \( x \). Let's break them down:

### Problem 32:
The function \( f(x) \) is given by:

\[
f(x) = 
\begin{cases} 
-x^2 & \text{if } x \leq -2 \\
3 & \text{if } -2 \leq x < 7 \\
(x - 5)^2 + 2 & \text{if } x \geq 7
\end{cases}
\]

- For \( x \leq -2 \), the function is a downward-facing parabola \( f(x) = -x^2 \).
- For \( -2 \leq x < 7 \), the function is a constant \( f(x) = 3 \).
- For \( x \geq 7 \), the function is a parabola \( f(x) = (x - 5)^2 + 2 \) shifted to the right by 5 and upwards by 2.

### Problem 33:
The function \( g(x) \) is given by:

\[
g(x) = 
\begin{cases} 
x + 4 & \text{if } x < -6 \\
\frac{1}{x} & \text{if } -6 \leq x < 4 \\
6 & \text{if } x \geq 4
\end{cases}
\]

- For \( x < -6 \), the function is a linear equation \( g(x) = x + 4 \).
- For \( -6 \leq x < 4 \), the function is the reciprocal \( g(x) = \frac{1}{x} \).
- For \( x \geq 4 \), the function is a constant \( g(x) = 6 \).

### Problem 34:
The function \( f(x) \) is given by:

\[
f(x) = 
\begin{cases} 
4 & \text{if } x < -5 \\
\frac{x^3}{\sqrt{x + 3}} & \text{if } -2 \leq x \leq 2 \\
\sqrt{x + 3} & \text{if } x > 3
\end{cases}
\]

- For \( x < -5 \), the function is constant \( f(x) = 4 \).
- For \( -2 \leq x \leq 2 \), the function is a rational function \( f(x) = \frac{x^3}{\sqrt{x + 3}} \).
- For \( x > 3 \), the function is the square root \( f(x) = \sqrt{x + 3} \).

Would you like me to graph these functions or go over any specific part in more detail?

### 5 Follow-up Questions:
1. How do you determine continuity in a piecewise function?
2. What are the critical points where the function changes in each piecewise case?
3. How would you evaluate \( f(x) \) for \( x = 7 \) in Problem 32?
4. What is the domain of the function in Problem 34?
5. Can you find the derivative of each segment in Problem 33?

### Tip:
When graphing piecewise functions, always evaluate the boundary points to check if they are included or excluded from each piece. This helps in drawing open or closed circles at transition points.

Graph each piecewise function f(x) and g(x) provided in problems 32, 33, and 34.

Learn how to graph complex piecewise functions that include parabolas, linear functions, and rational functions. This problem covers essential algebraic concepts and is suitable for high school students.

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