Evaluate 

*f*(−9),
 
*f*(0),

 and 

*f*(6)

 for the piecewise defined function.

*f*(*x*) = 

*x* + 2

    if *x* < 0

1 − *x*

    if *x* ≥ 0 find *f*(−9)


We are given a piecewise-defined function \( f(x) \), defined as follows:

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

We need to evaluate \( f(-9) \), \( f(0) \), and \( f(6) \).

### Evaluating \( f(-9) \):
Since \( -9 < 0 \), we use the first piece of the function:
\[
f(x) = x + 2.
\]
Substituting \( x = -9 \):
\[
f(-9) = -9 + 2 = -7.
\]

### Evaluating \( f(0) \):
Since \( 0 \geq 0 \), we use the second piece of the function:
\[
f(x) = 1 - x.
\]
Substituting \( x = 0 \):
\[
f(0) = 1 - 0 = 1.
\]

### Evaluating \( f(6) \):
Since \( 6 \geq 0 \), we use the second piece of the function:
\[
f(x) = 1 - x.
\]
Substituting \( x = 6 \):
\[
f(6) = 1 - 6 = -5.
\]

### Final Answers:
- \( f(-9) = -7 \)
- \( f(0) = 1 \)
- \( f(6) = -5 \)

Let me know if you would like any further clarification!

### Related Questions:
1. What is a piecewise function?
2. How do you determine which piece of a piecewise function to use?
3. Can a piecewise function be continuous at the point where the pieces meet?
4. How do you graph a piecewise function?
5. What is the significance of piecewise functions in real-life scenarios?

**Tip:** When working with piecewise functions, always check the domain condition carefully to determine which rule to apply for the given value of \( x \).

Learn how to evaluate the piecewise function f(x) = x + 2 if x < 0, 1 - x if x >= 0 at specific values such as f(-9), f(0), and f(6). Understand the application of piecewise functions in mathematical analysis with detailed step-by-step solutions.

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