https://file.aiquickdraw.com/mathbot/file/1727745419818ilowg8tm.pdf

Here is the first problem from the document:

### Problem 1:
You are given a piecewise function:
\[
f(x) =
\begin{cases}
2x + 3 & \text{if } x < -1 \\
-x - 2 & \text{if } x \geq -1
\end{cases}
\]
You are asked to find the value of \( f(-3) + f(5) \).

#### Solution:
- **Step 1**: Calculate \( f(-3) \).
  Since \( -3 < -1 \), we use the first part of the piecewise function \( f(x) = 2x + 3 \):
  \[
  f(-3) = 2(-3) + 3 = -6 + 3 = -3
  \]

- **Step 2**: Calculate \( f(5) \).
  Since \( 5 \geq -1 \), we use the second part of the piecewise function \( f(x) = -x - 2 \):
  \[
  f(5) = -5 - 2 = -7
  \]

- **Step 3**: Add the results.
  \[
  f(-3) + f(5) = -3 + (-7) = -10
  \]

Thus, the value of \( f(-3) + f(5) \) is **-10**.

Let me know if you want detailed explanations for other problems, or if you have any questions. Here are five related questions that can extend this topic:

1. How can you determine which part of a piecewise function to use based on the input value?
2. What are the key characteristics of piecewise functions?
3. How can we graph a piecewise function with different expressions on the same plot?
4. Can a piecewise function be continuous? If so, under what conditions?
5. What is the significance of the domain in defining a piecewise function?

**Tip**: When dealing with piecewise functions, always check the condition on \( x \) carefully to use the correct expression.

Find the value of f(-3) + f(5) for the piecewise function: f(x) = { 2x + 3 if x < -1; -x - 2 if x ≥ -1 }.

Learn how to evaluate the piecewise function f(x), which involves two cases: 2x + 3 for x < -1 and -x - 2 for x ≥ -1. This example calculates f(-3) and f(5), providing a step-by-step solution.

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