https://file.aiquickdraw.com/mathbot/file/1728435044518vl9wixs0.jpg

The image presents a piecewise function \( f(x) \) defined as follows:

\[
f(x) = 
\begin{cases} 
2x + 5 & \text{if } x < 0 \\
3x + 6 & \text{if } x \geq 0 
\end{cases}
\]

You are asked to evaluate \( f(x) \) at specific values of \( x \):
1. \( f(-1) \)
2. \( f(0) \)
3. \( f(4) \)

### Solutions:

1. **\( f(-1) \):**
   Since \( -1 < 0 \), use the first equation: \( f(x) = 2x + 5 \).
   \[
   f(-1) = 2(-1) + 5 = -2 + 5 = 3
   \]

2. **\( f(0) \):**
   Since \( 0 \geq 0 \), use the second equation: \( f(x) = 3x + 6 \).
   \[
   f(0) = 3(0) + 6 = 0 + 6 = 6
   \]

3. **\( f(4) \):**
   Since \( 4 \geq 0 \), use the second equation: \( f(x) = 3x + 6 \).
   \[
   f(4) = 3(4) + 6 = 12 + 6 = 18
   \]

### Final answers:
- \( f(-1) = 3 \)
- \( f(0) = 6 \)
- \( f(4) = 18 \)

Let me know if you need further explanations, or if you have any questions!

### Related questions:
1. What is \( f(x) \) if \( x = -2 \)?
2. How would the function behave if we added a third piece for \( x > 5 \)?
3. What is the general shape of the graph of this piecewise function?
4. Can you find the value of \( x \) when \( f(x) = 0 \)?
5. What is the slope of the line when \( x \geq 0 \)?

### Tip:
When working with piecewise functions, always check which condition \( x \) satisfies before applying the correct equation.

Evaluate the piecewise function at the given values of x for the function f(x) defined as:

f(x) = 
2x + 5 if x < 0
3x + 6 if x ≥ 0

a) f(-1)
b) f(0)
c) f(4)

Learn how to evaluate the piecewise function f(x) = 2x + 5 if x < 0, and f(x) = 3x + 6 if x ≥ 0, at specific values x = -1, x = 0, and x = 4. This step-by-step explanation demonstrates key algebraic concepts for Grades 9-12.

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