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The problem involves a piecewise function \( f(x) \) that is defined in three intervals, and we are tasked with finding the range of the function. Let's go through the function step by step:

### Function Definition:
\[
f(x) =
\begin{cases}
x + 1 & \text{if } x \leq -1 \\
x^2 - 1 & \text{if } -1 < x \leq 2 \\
-3x + 9 & \text{if } x > 2
\end{cases}
\]

### Analyzing Each Piece:

1. **For \( x \leq -1 \),** we have \( f(x) = x + 1 \):
   - The smallest value of \( x \) in this interval is \( x = -1 \), and at this point:
     \[
     f(-1) = -1 + 1 = 0
     \]
   - As \( x \to -\infty \), \( f(x) \to -\infty \).
   - So, for \( x \leq -1 \), \( f(x) \) takes values from \( (-\infty, 0] \).

2. **For \( -1 < x \leq 2 \),** we have \( f(x) = x^2 - 1 \):
   - This is a quadratic function that opens upwards, with the vertex at \( x = 0 \):
     \[
     f(0) = 0^2 - 1 = -1
     \]
   - The function increases as \( x \) moves away from 0. At \( x = 2 \):
     \[
     f(2) = 2^2 - 1 = 3
     \]
   - So, for \( -1 < x \leq 2 \), \( f(x) \) takes values from \( [-1, 3] \).

3. **For \( x > 2 \),** we have \( f(x) = -3x + 9 \):
   - This is a linear function with a negative slope, decreasing as \( x \) increases. At \( x = 2 \), this function would give:
     \[
     f(2) = -3(2) + 9 = 3
     \]
     but this value is already included in the second piece.
   - As \( x \to \infty \), \( f(x) \to -\infty \).
   - So, for \( x > 2 \), \( f(x) \) takes values from \( (-\infty, 3] \).

### Range of the Function:
- From the analysis of all three intervals, the range of \( f(x) \) combines the intervals:
  - \( (-\infty, 0] \) from the first piece,
  - \( [-1, 3] \) from the second piece,
  - and \( (-\infty, 3] \) from the third piece.
  
  Therefore, the range of the function is \( (-\infty, 3] \).

### Correct Answer:
The correct option is **e. \( \{ y \mid y \leq 3 \} \)**.

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Do you need further explanation or clarifications?

Here are some related questions for practice:
1. What is the domain of the function \( f(x) \)?
2. How would the range change if the middle interval was \( -1 < x \leq 3 \)?
3. Can you sketch the graph of \( f(x) \)?
4. What happens to the range if we add a constant to each piece of the function?
5. How do you determine the domain of a piecewise function?

**Tip:** Always examine the behavior of the function at boundary points when determining the range of a piecewise function.

Diketahui fungsi f(x) sebagai fungsi pecahan dengan tiga interval. Hitung daerah hasil fungsi.

Learn how to find the range of a piecewise function defined by linear and quadratic expressions in different intervals. This math problem involves understanding the behavior of functions across multiple intervals and determining the range.

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