Sketch the graph of the given function.
f(x) =

−x2	    	if −3 < x ≤ 0

x
if 0 < x < 9
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Evaluate the given expressions.
f(−1)	=
f(0)	=
f(1)	=

Let's start by evaluating the given piecewise function \( f(x) \) and determining \( f(-1) \), \( f(0) \), and \( f(1) \).

The function is defined as:

\[
f(x) = \begin{cases} 
-x^2 & \text{if } -3 < x \leq 0 \\
x & \text{if } 0 < x < 9
\end{cases}
\]

### Evaluations:

1. **\( f(-1) \):**
   Since \( -1 \) is in the range \( -3 < x \leq 0 \), we use the first piece of the function:
   \[
   f(-1) = -(-1)^2 = -1
   \]

2. **\( f(0) \):**
   Since \( 0 \) is in the range \( -3 < x \leq 0 \), we use the first piece of the function:
   \[
   f(0) = -(0)^2 = 0
   \]

3. **\( f(1) \):**
   Since \( 1 \) is in the range \( 0 < x < 9 \), we use the second piece of the function:
   \[
   f(1) = 1
   \]

### Final answers:
- \( f(-1) = -1 \)
- \( f(0) = 0 \)
- \( f(1) = 1 \)

Would you like a sketch of the graph of this function? Feel free to ask any follow-up questions.

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Here are 5 related questions to expand on this problem:

1. How would the graph of \( f(x) \) change if the first piece were \( x^2 \) instead of \( -x^2 \)?
2. What is the behavior of \( f(x) \) as \( x \to -3^+ \) and \( x \to 9^- \)?
3. How would you describe the continuity of this piecewise function?
4. What would happen if the function had an additional piece for \( x \geq 9 \)?
5. How can we determine whether the function is differentiable at \( x = 0 \)?

**Tip:** When evaluating a piecewise function, always make sure to check which part of the domain the input belongs to.

Sketch the graph of the given function. f(x) = -x^2 if -3 < x ≤ 0, x if 0 < x < 9. Evaluate the given expressions: f(−1) = ?, f(0) = ?, f(1) = ?

Learn how to graph and evaluate a piecewise function where f(x) is defined as -x^2 for -3 < x ≤ 0 and x for 0 < x < 9. The solution includes step-by-step evaluations of f(−1), f(0), and f(1), making this problem ideal for students in Grades 9-12.

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