Worksheet for graphing piecewise functions with corresponding equations and blank coordinate planes.

Let's solve each of these piecewise functions step by step. For each function, we will:

1. Graph the function on the given coordinate plane.
2. Determine the domain and range.

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Problem 1:

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

#### Step-by-step:
- For $ x \leq 2 $: $ f(x) = -x $. This is a line with slope $-1$. It includes $ x = 2 $, so use a closed circle at $ (2, -2) $.
- For $ x > 2 $: $ f(x) = x $. This is a line with slope $1$. Since it's $ x > 2 $, use an open circle at $ (2, 2) $, then draw the ray to the right.

#### Graph:
- Left part: Line $ y = -x $ from left to $ (2, -2) $, closed circle.
- Right part: Line $ y = x $ starting just after $ x = 2 $, open circle at $ (2, 2) $, going up and right.

#### Domain:
All real numbers: $ (-\infty, \infty) $

#### Range:
- For $ x \leq 2 $: $ f(x) = -x $ → as $ x \to -\infty $, $ f(x) \to \infty $; at $ x=2 $, $ f(2) = -2 $
So this part gives values $ [-2, \infty) $
- For $ x > 2 $: $ f(x) = x $ → starts above $ x=2 $, so $ f(x) > 2 $
So this part gives $ (2, \infty) $

But note: $ [-2, \infty) $ already includes $ (2, \infty) $, so overall range is $ [-2, \infty) $

✔ Domain: $ (-\infty, \infty) $
✔ Range: $ [-2, \infty) $

---

Problem 2:

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

Note: There's no definition at $ x = -3 $. So the function is undefined at $ x = -3 $.

- For $ x > -3 $: constant value $ f(x) = 2 $
- For $ x < -3 $: constant value $ f(x) = -5 $

#### Graph:
- Draw horizontal line $ y = 2 $ for $ x > -3 $ → open circle at $ (-3, 2) $
- Draw horizontal line $ y = -5 $ for $ x < -3 $ → open circle at $ (-3, -5) $
- No point at $ x = -3 $

#### Domain:
All real numbers except $ x = -3 $ → $ (-\infty, -3) \cup (-3, \infty) $

#### Range:
Only two values: $ -5 $ and $ 2 $

✔ Domain: $ (-\infty, -3) \cup (-3, \infty) $
✔ Range: $ \{-5, 2\} $

---

Problem 3:

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

- For $ x \leq -2 $: constant $ f(x) = -1 $
- For $ x > -2 $: constant $ f(x) = 2 $

#### Graph:
- Horizontal line $ y = -1 $ from left to $ x = -2 $, closed circle at $ (-2, -1) $
- Horizontal line $ y = 2 $ for $ x > -2 $, open circle at $ (-2, 2) $

#### Domain:
All real numbers: $ (-\infty, \infty) $

#### Range:
Values are $ -1 $ and $ 2 $

✔ Domain: $ (-\infty, \infty) $
✔ Range: $ \{-1, 2\} $

---

Problem 4:

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

Break into three parts:

1. $ x \leq -1 $: $ f(x) = -1 $ → horizontal line at $ y = -1 $, closed circle at $ (-1, -1) $
2. $ -1 < x < 1 $: $ f(x) = 1 $ → horizontal line at $ y = 1 $, open circles at both ends: $ (-1, 1) $ and $ (1, 1) $
3. $ x > 1 $: $ f(x) = x $ → line $ y = x $, open circle at $ (1, 1) $, then increasing

#### Graph:
- Left: flat line $ y = -1 $ ending at $ x = -1 $, closed
- Middle: flat line $ y = 1 $ from $ x = -1 $ to $ x = 1 $, open at both ends
- Right: line $ y = x $ starting after $ x = 1 $, open at $ (1,1) $

#### Domain:
All real numbers: $ (-\infty, \infty) $

#### Range:
- From first part: $ y = -1 $
- Second: $ y = 1 $
- Third: $ y = x $ for $ x > 1 $ → $ y > 1 $

So total range: $ \{-1\} \cup \{1\} \cup (1, \infty) = \{-1\} \cup [1, \infty) $

Wait: $ y = 1 $ is included in middle, and $ y > 1 $ from third, so together $ [1, \infty) $

Thus: $ \{-1\} \cup [1, \infty) $

✔ Domain: $ (-\infty, \infty) $
✔ Range: $ \{-1\} \cup [1, \infty) $

---

Problem 5:

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

#### Part 1: $ x \leq 0 $, $ f(x) = -x + 2 $
- Slope: $-1$, y-intercept: $2$
- At $ x = 0 $: $ f(0) = -0 + 2 = 2 $ → closed circle at $ (0, 2) $
- Draw line to the left

#### Part 2: $ x > 0 $, $ f(x) = \frac{1}{2}x + 3 $
- Slope: $ \frac{1}{2} $, y-intercept: $3$
- But only for $ x > 0 $, so open circle at $ (0, 3) $
- Then draw line upward to the right

#### Graph:
- Left: line $ y = -x + 2 $ from left to $ (0, 2) $, closed
- Right: line $ y = \frac{1}{2}x + 3 $ starting at $ (0, 3) $, open circle, going up

#### Domain:
All real numbers: $ (-\infty, \infty) $

#### Range:
- Left part: $ x \leq 0 $, $ f(x) = -x + 2 $
- As $ x \to -\infty $, $ f(x) \to \infty $
- At $ x = 0 $, $ f(0) = 2 $
→ So this part gives $ [2, \infty) $
- Right part: $ x > 0 $, $ f(x) = \frac{1}{2}x + 3 $
- At $ x \to 0^+ $, $ f(x) \to 3 $
- As $ x \to \infty $, $ f(x) \to \infty $
→ So this part gives $ (3, \infty) $

Now combine:
- $ [2, \infty) $ from left
- $ (3, \infty) $ from right

But $ [2, \infty) $ already includes $ (3, \infty) $, and also includes $ [2, 3] $

So overall range: $ [2, \infty) $

✔ Domain: $ (-\infty, \infty) $
✔ Range: $ [2, \infty) $

---

Problem 6:

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

#### Part 1: $ x \leq 2 $, $ f(x) = x + 2 $
- Slope: $1$, y-intercept: $2$
- At $ x = 2 $: $ f(2) = 2 + 2 = 4 $ → closed circle at $ (2, 4) $
- Draw line to the left

#### Part 2: $ x > 2 $, $ f(x) = -\frac{1}{2}x + 4 $
- Slope: $-\frac{1}{2}$, y-intercept: $4$
- At $ x = 2 $: $ f(2) = -\frac{1}{2}(2) + 4 = -1 + 4 = 3 $, but since $ x > 2 $, open circle at $ (2, 3) $
- Draw decreasing line to the right

#### Graph:
- Left: line $ y = x + 2 $ up to $ (2, 4) $, closed
- Right: line $ y = -\frac{1}{2}x + 4 $ starting after $ x = 2 $, open at $ (2, 3) $, decreasing

#### Domain:
All real numbers: $ (-\infty, \infty) $

#### Range:
- Left part: $ x \leq 2 $, $ f(x) = x + 2 $
- As $ x \to -\infty $, $ f(x) \to -\infty $
- At $ x = 2 $, $ f(2) = 4 $
→ So this part gives $ (-\infty, 4] $
- Right part: $ x > 2 $, $ f(x) = -\frac{1}{2}x + 4 $
- As $ x \to 2^+ $, $ f(x) \to 3 $
- As $ x \to \infty $, $ f(x) \to -\infty $
→ So this part gives $ (-\infty, 3) $

Now combine:
- Left: $ (-\infty, 4] $
- Right: $ (-\infty, 3) $

Union: $ (-\infty, 4] $

✔ Domain: $ (-\infty, \infty) $
✔ Range: $ (-\infty, 4] $

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✔ Summary Table:

| Problem | Domain | Range |
|--------|--------|-------|
| 1 | $ (-\infty, \infty) $ | $ [-2, \infty) $ |
| 2 | $ (-\infty, -3) \cup (-3, \infty) $ | $ \{-5, 2\} $ |
| 3 | $ (-\infty, \infty) $ | $ \{-1, 2\} $ |
| 4 | $ (-\infty, \infty) $ | $ \{-1\} \cup [1, \infty) $ |
| 5 | $ (-\infty, \infty) $ | $ [2, \infty) $ |
| 6 | $ (-\infty, \infty) $ | $ (-\infty, 4] $ |

You can now graph each piece using the descriptions above. Make sure to:
- Use closed circles where the inequality includes equality (≤ or ≥)
- Use open circles where it does not ( < or > )
- Connect appropriately with lines or rays

Let me know if you'd like visual sketches or further explanation!