Linear Piecewise Functions Worksheet - Fill Online, Printable ... - Free Printable

Let's solve each of the piecewise functions and explain how to graph them step by step. I'll go through each problem, describe the function, determine the correct graphing method, and provide a clear explanation.

---

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 $: Graph $ y = -x $. This is a line with slope $-1$, passing through points like $ (0,0), (2,-2), (-2,2) $.
- Since $ x \leq 2 $, include the point at $ x = 2 $: $ f(2) = -2 $. Use a closed circle at $ (2, -2) $.
- For $ x > 2 $: Graph $ y = x $. This is a line with slope $1$, starting just after $ x = 2 $.
- At $ x = 2 $, $ f(x) = x $ would be $ 2 $, but since $ x > 2 $, we do not include $ x = 2 $ here.
- Use an open circle at $ (2, 2) $, then draw the line $ y = x $ for $ x > 2 $.

#### Graph Description:
- Left side: Line $ y = -x $ from left up to $ (2, -2) $ (closed).
- Right side: Line $ y = x $ starting from $ (2, 2) $ (open) going upward.

---

2.

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

Wait — this function has no definition at $ x = -3 $. Let’s check:

- $ x > -3 $ → $ f(x) = 2 $
- $ x < -3 $ → $ f(x) = -5 $

But nothing is defined at $ x = -3 $. That’s okay — we can still graph it.

#### Step-by-step:
- For $ x < -3 $: Constant function $ f(x) = -5 $. Draw a horizontal line at $ y = -5 $ for all $ x < -3 $.
- Use an open circle at $ x = -3 $ because $ x < -3 $ does not include $ x = -3 $.
- For $ x > -3 $: Constant function $ f(x) = 2 $. Draw a horizontal line at $ y = 2 $ for all $ x > -3 $.
- Use an open circle at $ x = -3 $ here too, since $ x > -3 $ doesn't include $ x = -3 $.

At $ x = -3 $, there is a gap — no value defined.

#### Graph Description:
- Two horizontal lines:
- $ y = -5 $ for $ x < -3 $, open circle at $ (-3, -5) $
- $ y = 2 $ for $ x > -3 $, open circle at $ (-3, 2) $
- No point at $ x = -3 $

---

3.

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

#### Step-by-step:
- For $ x \leq -2 $: $ f(x) = -1 $. Horizontal line at $ y = -1 $, including $ x = -2 $ → closed circle at $ (-2, -1) $.
- For $ x > -2 $: $ f(x) = 2 $. Horizontal line at $ y = 2 $, starting just after $ x = -2 $ → open circle at $ (-2, 2) $.

#### Graph Description:
- Left: Horizontal line $ y = -1 $ for $ x \leq -2 $, closed at $ (-2, -1) $
- Right: Horizontal line $ y = 2 $ for $ x > -2 $, open at $ (-2, 2) $

---

4.

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

#### Step-by-step:
- For $ x \leq -1 $: $ f(x) = -1 $ → horizontal line at $ y = -1 $, closed at $ x = -1 $ → closed circle at $ (-1, -1) $
- For $ -1 < x < 1 $: $ f(x) = 1 $ → horizontal line at $ y = 1 $, but not including endpoints.
- Open circles at $ x = -1 $ and $ x = 1 $
- For $ x > 1 $: $ f(x) = x $ → line with slope 1, starting just after $ x = 1 $
- Open circle at $ (1, 1) $, then draw $ y = x $ for $ x > 1 $

#### Graph Description:
- Left: $ y = -1 $ for $ x \leq -1 $, closed at $ (-1, -1) $
- Middle: $ y = 1 $ between $ x = -1 $ and $ x = 1 $, open circles at both ends
- Right: $ y = x $ for $ x > 1 $, open at $ (1, 1) $

---

5.

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

#### Step-by-step:
- For $ x \leq 0 $: $ f(x) = -x + 2 $
- This is a line with slope $-1$, y-intercept $2$
- At $ x = 0 $: $ f(0) = -0 + 2 = 2 $ → closed circle at $ (0, 2) $
- Points: $ (0,2), (-2,4), (-4,6) $
- For $ x > 0 $: $ f(x) = \frac{1}{2}x + 3 $
- Slope $ \frac{1}{2} $, y-intercept $3$
- But starts at $ x > 0 $, so open circle at $ (0, 3) $
- Then plot points: $ (2, 4), (4, 5) $, etc.

Note: At $ x = 0 $, the function uses the first rule: $ f(0) = 2 $, not $ \frac{1}{2}(0)+3 = 3 $

#### Graph Description:
- Left: Line $ y = -x + 2 $ for $ x \leq 0 $, closed at $ (0,2) $
- Right: Line $ y = \frac{1}{2}x + 3 $ for $ x > 0 $, open at $ (0,3) $

---

6.

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

#### Step-by-step:
- For $ 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) $
- For $ 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 not included, so open circle at $ (2, 3) $
- Then plot points: $ (4, 2), (6, 1) $, etc.

#### Graph Description:
- Left: Line $ y = x + 2 $ for $ x \leq 2 $, closed at $ (2,4) $
- Right: Line $ y = -\frac{1}{2}x + 4 $ for $ x > 2 $, open at $ (2,3) $

---

✔ Summary of Key Graphing Rules:

| Condition | Circle Type |
|---------|-------------|
| $ x \leq a $ or $ x \geq a $ | Closed circle at $ x = a $ |
| $ x < a $ or $ x > a $ | Open circle at $ x = a $ |

Always:
- Graph each piece separately.
- Pay attention to domain restrictions.
- Label key points and use proper circle types.

---

If you'd like, I can generate a sketch of one or more graphs in text form (ASCII art), or help you understand how to plot specific points. Let me know!