Give the domain and range.​ Then, use the graph to find ​(​a)​ f(negative 2​), ​(​b)​ f(0​), ​(​c) f left parenthesis one half right parenthesis comma
 and ​(​d) any values of x such that ​f(x)equals2.
-2
2
4
-2
2
4
6
x
y



A coordinate system has a horizontal x-axis labeled from negative 3 to 5 in increments of 1 and a vertical y-axis labeled from negative 2 to 6 in increments of 1. The graph consists of three line segments. A line segment rises from left to right from the closed plotted point (negative 2, 0) to the point (0, 4). Another line segment falls from left to right from the point (0, 4) to the point (2, 0). Another line segment rises from left to right from the point (2, 0) to the closed plotted point (4, 4).
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Question content area right
Part 1
The domain is 
  
enter your response here.

### Part 1: Domain and Range
**Domain:** The domain consists of all \( x \)-values where the graph is defined. Based on the description, the graph starts at \( x = -2 \) and ends at \( x = 4 \), including both endpoints.

Thus, the domain is:
\[
\boxed{[-2, 4]}
\]

**Range:** The range consists of all \( y \)-values where the graph is defined. From the description, the lowest \( y \)-value is \( 0 \) (at \( x = -2 \) and \( x = 2 \)) and the highest \( y \)-value is \( 4 \) (at \( x = 0 \) and \( x = 4 \)).

Thus, the range is:
\[
\boxed{[0, 4]}
\]

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### Part 2: Evaluate the function
Using the graph to find the corresponding function values:

- **(a) \( f(-2) \):** From the graph, at \( x = -2 \), \( y = 0 \).
  \[
  f(-2) = \boxed{0}
  \]

- **(b) \( f(0) \):** From the graph, at \( x = 0 \), \( y = 4 \).
  \[
  f(0) = \boxed{4}
  \]

- **(c) \( f\left(\frac{1}{2}\right) \):** Since \( x = \frac{1}{2} \) lies on the line segment between \( x = 0 \) and \( x = 2 \), we can interpolate. The line segment falls from \( (0, 4) \) to \( (2, 0) \). The slope is:
  \[
  \text{slope} = \frac{0 - 4}{2 - 0} = -2
  \]
  So, for \( x = \frac{1}{2} \):
  \[
  f\left(\frac{1}{2}\right) = 4 + (-2) \times \frac{1}{2} = 4 - 1 = \boxed{3}
  \]

- **(d) Values of \( x \) such that \( f(x) = 2 \):**
  From the graph, \( f(x) = 2 \) occurs at two points:
  - On the increasing segment from \( (-2, 0) \) to \( (0, 4) \), it occurs when \( x = -1 \).
  - On the decreasing segment from \( (0, 4) \) to \( (2, 0) \), it occurs when \( x = 1 \).

Thus, \( f(x) = 2 \) at \( x = \boxed{-1 \text{ and } 1} \).

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Let me know if you need any further explanation!

### Related Questions:
1. How do we calculate the slope of a line segment between two points?
2. How can we use interpolation to estimate values on a line segment?
3. What is the difference between the domain and the range of a function?
4. How can we identify the values of \( x \) for a given \( f(x) \)?
5. What is the significance of closed points in a graph?

**Tip:** When analyzing piecewise functions, always break down the function into its segments to properly evaluate each part!

Give the domain and range. Then, use the graph to find (a) f(negative 2), (b) f(0), (c) f(1/2), and (d) any values of x such that f(x) = 2. A coordinate system has a horizontal x-axis labeled from -3 to 5 and a vertical y-axis labeled from -2 to 6. The graph consists of three line segments connecting points (-2, 0), (0, 4), (2, 0), and (4, 4).

Analyze a piecewise function and its graph to determine the domain and range. Learn how to evaluate the function at specific points such as f(-2), f(0), and f(1/2), and find values of x where f(x) equals 2. Suitable for Grades 8-10.

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