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Problem: Graphing Piecewise Functions

The task is to graph the given piecewise functions on the provided coordinate grids. Below, I will solve each problem step by step and explain the process.

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Problem 1:

$$
f(x) =
\begin{cases}
-x & \text{if } x \leq 2 \\
x & \text{if } x > 2
\end{cases}
$$

#### Steps:
1. Identify the two pieces of the function:
- For \( x \leq 2 \): \( f(x) = -x \)
- For \( x > 2 \): \( f(x) = x \)

2. Graph \( f(x) = -x \) for \( x \leq 2 \):
- This is a straight line with a slope of \(-1\) and passing through the origin.
- Plot points such as \((0, 0)\), \((1, -1)\), \((2, -2)\).
- Since \( x \leq 2 \), the line extends to the left and includes the point \((2, -2)\). Use a closed circle at \((2, -2)\).

3. Graph \( f(x) = x \) for \( x > 2 \):
- This is a straight line with a slope of \(1\) and passing through the origin.
- Plot points such as \((3, 3)\), \((4, 4)\).
- Since \( x > 2 \), the line starts just after \( x = 2 \). Use an open circle at \((2, 2)\).

#### Final Graph:
- A line \( y = -x \) for \( x \leq 2 \) with a closed circle at \((2, -2)\).
- A line \( y = x \) for \( x > 2 \) with an open circle at \((2, 2)\).

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Problem 2:

$$
f(x) =
\begin{cases}
2 & \text{if } x > -3 \\
-5 & \text{if } x < -3
\end{cases}
$$

#### Steps:
1. Identify the two pieces of the function:
- For \( x > -3 \): \( f(x) = 2 \)
- For \( x < -3 \): \( f(x) = -5 \)

2. Graph \( f(x) = 2 \) for \( x > -3 \):
- This is a horizontal line at \( y = 2 \).
- Since \( x > -3 \), the line starts just after \( x = -3 \). Use an open circle at \((-3, 2)\).

3. Graph \( f(x) = -5 \) for \( x < -3 \):
- This is a horizontal line at \( y = -5 \).
- Since \( x < -3 \), the line extends to the left and includes all points before \( x = -3 \). Use a closed circle at \((-3, -5)\).

#### Final Graph:
- A horizontal line \( y = 2 \) for \( x > -3 \) with an open circle at \((-3, 2)\).
- A horizontal line \( y = -5 \) for \( x < -3 \) with a closed circle at \((-3, -5)\).

---

Problem 3:

$$
f(x) =
\begin{cases}
-1 & \text{if } x \leq -2 \\
2 & \text{if } x > -2
\end{cases}
$$

#### Steps:
1. Identify the two pieces of the function:
- For \( x \leq -2 \): \( f(x) = -1 \)
- For \( x > -2 \): \( f(x) = 2 \)

2. Graph \( f(x) = -1 \) for \( x \leq -2 \):
- This is a horizontal line at \( y = -1 \).
- Since \( x \leq -2 \), the line includes all points up to and including \( x = -2 \). Use a closed circle at \((-2, -1)\).

3. Graph \( f(x) = 2 \) for \( x > -2 \):
- This is a horizontal line at \( y = 2 \).
- Since \( x > -2 \), the line starts just after \( x = -2 \). Use an open circle at \((-2, 2)\).

#### Final Graph:
- A horizontal line \( y = -1 \) for \( x \leq -2 \) with a closed circle at \((-2, -1)\).
- A horizontal line \( y = 2 \) for \( x > -2 \) with an open circle at \((-2, 2)\).

---

Problem 4:

$$
f(x) =
\begin{cases}
-1 & \text{if } x \leq -1 \\
1 & \text{if } -1 < x < 1 \\
x & \text{if } x > 1
\end{cases}
$$

#### Steps:
1. Identify the three pieces of the function:
- For \( x \leq -1 \): \( f(x) = -1 \)
- For \( -1 < x < 1 \): \( f(x) = 1 \)
- For \( x > 1 \): \( f(x) = x \)

2. Graph \( f(x) = -1 \) for \( x \leq -1 \):
- This is a horizontal line at \( y = -1 \).
- Since \( x \leq -1 \), the line includes all points up to and including \( x = -1 \). Use a closed circle at \((-1, -1)\).

3. Graph \( f(x) = 1 \) for \( -1 < x < 1 \):
- This is a horizontal line at \( y = 1 \).
- Since \( -1 < x < 1 \), the line does not include \( x = -1 \) or \( x = 1 \). Use open circles at \((-1, 1)\) and \((1, 1)\).

4. Graph \( f(x) = x \) for \( x > 1 \):
- This is a straight line with a slope of \(1\) and passing through the origin.
- Since \( x > 1 \), the line starts just after \( x = 1 \). Use an open circle at \((1, 1)\).

#### Final Graph:
- A horizontal line \( y = -1 \) for \( x \leq -1 \) with a closed circle at \((-1, -1)\).
- A horizontal line \( y = 1 \) for \( -1 < x < 1 \) with open circles at \((-1, 1)\) and \((1, 1)\).
- A line \( y = x \) for \( x > 1 \) with an open circle at \((1, 1)\).

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Problem 5:

$$
f(x) =
\begin{cases}
-x + 2 & \text{if } x \leq 0 \\
\frac{1}{2}x + 3 & \text{if } x > 0
\end{cases}
$$

#### Steps:
1. Identify the two pieces of the function:
- For \( x \leq 0 \): \( f(x) = -x + 2 \)
- For \( x > 0 \): \( f(x) = \frac{1}{2}x + 3 \)

2. Graph \( f(x) = -x + 2 \) for \( x \leq 0 \):
- This is a straight line with a slope of \(-1\) and a y-intercept of \(2\).
- Plot points such as \((0, 2)\), \((-1, 3)\).
- Since \( x \leq 0 \), the line includes all points up to and including \( x = 0 \). Use a closed circle at \((0, 2)\).

3. Graph \( f(x) = \frac{1}{2}x + 3 \) for \( x > 0 \):
- This is a straight line with a slope of \(\frac{1}{2}\) and a y-intercept of \(3\).
- Plot points such as \((0, 3)\), \((2, 4)\).
- Since \( x > 0 \), the line starts just after \( x = 0 \). Use an open circle at \((0, 3)\).

#### Final Graph:
- A line \( y = -x + 2 \) for \( x \leq 0 \) with a closed circle at \((0, 2)\).
- A line \( y = \frac{1}{2}x + 3 \) for \( x > 0 \) with an open circle at \((0, 3)\).

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Problem 6:

$$
f(x) =
\begin{cases}
x + 2 & \text{if } x \leq 2 \\
-\frac{1}{2}x + 4 & \text{if } x > 2
\end{cases}
$$

#### Steps:
1. Identify the two pieces of the function:
- For \( x \leq 2 \): \( f(x) = x + 2 \)
- For \( x > 2 \): \( f(x) = -\frac{1}{2}x + 4 \)

2. Graph \( f(x) = x + 2 \) for \( x \leq 2 \):
- This is a straight line with a slope of \(1\) and a y-intercept of \(2\).
- Plot points such as \((0, 2)\), \((2, 4)\).
- Since \( x \leq 2 \), the line includes all points up to and including \( x = 2 \). Use a closed circle at \((2, 4)\).

3. Graph \( f(x) = -\frac{1}{2}x + 4 \) for \( x > 2 \):
- This is a straight line with a slope of \(-\frac{1}{2}\) and a y-intercept of \(4\).
- Plot points such as \((2, 3)\), \((4, 2)\).
- Since \( x > 2 \), the line starts just after \( x = 2 \). Use an open circle at \((2, 3)\).

#### Final Graph:
- A line \( y = x + 2 \) for \( x \leq 2 \) with a closed circle at \((2, 4)\).
- A line \( y = -\frac{1}{2}x + 4 \) for \( x > 2 \) with an open circle at \((2, 3)\).

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Final Answers:

The graphs for each problem are described above. The final boxed answer is:

\[
\boxed{\text{Graphs as described in the solutions above.}}
\]