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find domain and range, find increase, decrease, and constant, find zero of f

Let's break down the analysis of the graph provided for \( y = f(x) \).

### 1. Domain:
The domain is the set of all \( x \)-values for which the function is defined. By observing the graph:
- The function starts at \( x = -4 \) (closed circle, included point), and ends at \( x = 5 \) (open circle, excluded point).
Thus, the **domain** is:
\[
\text{Domain: } [-4, 5)
\]

### 2. Range:
The range is the set of all possible \( y \)-values of the function. From the graph:
- The minimum \( y \)-value is \( 0 \), occurring at \( x \approx 4.5 \), and the maximum \( y \)-value is \( 8 \), occurring at \( x = -2 \).
Thus, the **range** is:
\[
\text{Range: } [0, 8]
\]

### 3. Intervals of Increase, Decrease, and Constant:
- **Increasing**: The function is increasing where the slope of the curve is positive.
    - From \( x = -4 \) to \( x = -2 \), the function is increasing.
    - From \( x \approx 3 \) to \( x = 5 \), the function is also increasing.
    - **Increasing intervals**: \( (-4, -2) \cup (3, 5) \)
  
- **Decreasing**: The function is decreasing where the slope of the curve is negative.
    - From \( x = -2 \) to \( x = 3 \), the function is decreasing.
    - **Decreasing interval**: \( (-2, 3) \)

- **Constant**: There is no section where the function is constant.

### 4. Zeros of \( f(x) \) (where \( f(x) = 0 \)):
The zero of the function is where the curve intersects the x-axis (i.e., where \( y = 0 \)).
- From the graph, the zero is located at \( x = -3 \).
Thus, the **zero** is:
\[
\text{Zero: } x = -3
\]

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Would you like to dive deeper into any specific part of this problem?

Here are some related questions:
1. How would we find the critical points of this function?
2. Can we approximate the derivative of \( f(x) \) based on the graph?
3. How do we describe the concavity of the graph?
4. How can we use limits to describe the behavior of the function near \( x = 5 \)?
5. How could the function be modeled algebraically based on the graph?

**Tip**: When analyzing functions graphically, always check for open and closed endpoints to determine whether certain values are included in the domain or range.

Find domain and range, find increase, decrease, and constant, find zero of f.

Learn how to find the domain and range, intervals of increase and decrease, and zeros of the function y = f(x) from its graph. This problem covers essential mathematical concepts in functions, suitable for students in Grades 9-12.

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