The graph of a​ piecewise-defined function is given. Write a definition for the function that best describes this graph. 


-7
7
-7
7
x
y




left parenthesis 2 comma 1 right parenthesis
(2,1)

left parenthesis 0 comma 0 right parenthesis
(0,0)
left parenthesis negative 2 comma 2 right parenthesis
(−2,2)





A coordinate system has a horizontal x-axis labeled from negative 7 to 7 in increments of 1 and a vertical y-axis labeled from negative 7 to 7 in increments of 1. The graph contains two branches. The first branch is a ray that starts at a labeled solid circle at (0, 0), rises from right to left, and passes through the plotted and labeled point (negative 2, 2). The second branch is a line segment that falls from left to right and extends from an open circle at (0, 3), through the plotted and labeled point (2, 2), to a solid circle at (3, 0).

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Part 1

f left parenthesis x right parenthesis equals
f(x)=left brace Start 2 By 2 Matrix 1st Row 1st Column nothing 2nd Column if x less than or equals nothing 2nd Row 1st Column nothing 2nd Column if 0 less than x less than or equals nothing EndMatrix . please note that the points on the graph are as follows (-2,2), (0,0) and (2,1)

To define the piecewise function based on the given graph, we will break it into two parts:

1. **First branch**: This branch is a ray that starts from the point \((0, 0)\) and rises to the left, passing through the point \((-2, 2)\). This suggests a line with a negative slope. We can determine the equation of the line passing through these points.

2. **Second branch**: This is a line segment that starts from an open circle at \((0, 3)\) and ends at a closed circle at \((3, 0)\), passing through the point \((2, 1)\). This is a line segment with a negative slope.

### Finding the equations of each branch:

#### First Branch (for \(x \leq 0\)):

- Points on the line: \((0, 0)\) and \((-2, 2)\).
- Slope of the line: 
  \[
  m = \frac{2 - 0}{-2 - 0} = -1
  \]
- Equation of the line: Using point-slope form \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is one of the points on the line:
  \[
  y - 0 = -1(x - 0) \quad \Rightarrow \quad y = -x
  \]
  So, for \(x \leq 0\), the equation is \(f(x) = -x\).

#### Second Branch (for \(0 < x \leq 3\)):

- Points on the line: \((0, 3)\) and \((3, 0)\).
- Slope of the line:
  \[
  m = \frac{0 - 3}{3 - 0} = -1
  \]
- Equation of the line: Using the point-slope form again:
  \[
  y - 3 = -1(x - 0) \quad \Rightarrow \quad y = -x + 3
  \]
  So, for \(0 < x \leq 3\), the equation is \(f(x) = -x + 3\).

### The piecewise function is:

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

This function describes the two branches of the graph.

Would you like further details or have any questions?

---

Here are 5 questions to further your understanding:
1. How would the function change if the slope of the first branch was steeper?
2. Can you verify the continuity of the piecewise function at \(x = 0\)?
3. How would the function behave if the second branch extended beyond \(x = 3\)?
4. What if the open circle at \((0, 3)\) were a closed circle? How would that affect the function?
5. Can you find the domain and range of this piecewise function?

**Tip**: Always check if the piecewise function is continuous at the boundaries (e.g., at \(x = 0\) in this case).

The graph of a​ piecewise-defined function is given. Write a definition for the function that best describes this graph. Points on the graph are (-2,2), (0,0), and (2,1).

Learn how to define a piecewise function based on a graph with points (-2,2), (0,0), and (2,1). This problem covers piecewise functions, linear equations, and using the slope formula to determine equations of each segment. Suitable for high school students (Grades 9-12).

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