Piecewise Functions Practice Worksheet with Graphing and Evaluation Tasks

Let's solve each part of this piecewise function practice step by step.

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Problem 1:

Given:
$$
f(x) =
\begin{cases}
x + 5 & \text{if } x < -2 \\
-2x - 1 & \text{if } x \geq -2
\end{cases}
$$

We are to:
1. Graph the function.
2. Determine if it is a function (it will be, since it's piecewise-defined with no contradictions).
3. Evaluate $ f(3), f(-4), f(-2) $.

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#### Step 1: Evaluate the function at given values

- $ f(3) $:
Since $ 3 \geq -2 $, use the second rule: $ f(x) = -2x - 1 $
$$
f(3) = -2(3) - 1 = -6 - 1 = -7
$$

- $ f(-4) $:
Since $ -4 < -2 $, use the first rule: $ f(x) = x + 5 $
$$
f(-4) = -4 + 5 = 1
$$

- $ f(-2) $:
Since $ -2 \geq -2 $, use the second rule: $ f(x) = -2x - 1 $
$$
f(-2) = -2(-2) - 1 = 4 - 1 = 3
$$

✔ So:
- $ f(3) = -7 $
- $ f(-4) = 1 $
- $ f(-2) = 3 $

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#### Graphing Instructions (for Problem 1):

We have two pieces:

1. For $ x < -2 $: $ y = x + 5 $
- This is a straight line with slope 1 and y-intercept 5.
- But only for $ x < -2 $. At $ x = -2 $, we do not include this part.
- So draw a line starting from left, ending with an open circle at $ x = -2 $.

Let’s find a few points:
- $ x = -3 $: $ y = -3 + 5 = 2 $ → point $ (-3, 2) $
- $ x = -4 $: $ y = -4 + 5 = 1 $ → point $ (-4, 1) $
- $ x = -5 $: $ y = 0 $

Plot these and draw a ray going to the left with open circle at $ x = -2 $.

2. For $ x \geq -2 $: $ y = -2x - 1 $
- Slope = -2, y-intercept = -1.
- Includes $ x = -2 $, so closed circle at that point.

Find a few points:
- $ x = -2 $: $ y = -2(-2) - 1 = 4 - 1 = 3 $ → point $ (-2, 3) $
- $ x = 0 $: $ y = -1 $
- $ x = 1 $: $ y = -2(1) - 1 = -3 $
- $ x = 3 $: $ y = -7 $

Draw a ray starting at $ (-2, 3) $ with closed circle and continue to the right.

✔ The graph consists of two rays meeting at $ x = -2 $, but only one side includes the point.

✔ Is this a function?
Yes — each input has exactly one output. Passes vertical line test.

---

Problem 2:

Given:
$$
f(x) =
\begin{cases}
2x + 1 & \text{if } x \geq 1 \\
\frac{x}{2} - 3 & \text{if } x < 1
\end{cases}
$$

Evaluate:
- $ f(-2) $
- $ f(6) $
- $ f(1) $

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#### Step 1: Evaluate the function

- $ f(-2) $:
Since $ -2 < 1 $, use $ f(x) = \frac{x}{2} - 3 $
$$
f(-2) = \frac{-2}{2} - 3 = -1 - 3 = -4
$$

- $ f(6) $:
Since $ 6 \geq 1 $, use $ f(x) = 2x + 1 $
$$
f(6) = 2(6) + 1 = 12 + 1 = 13
$$

- $ f(1) $:
Since $ 1 \geq 1 $, use $ f(x) = 2x + 1 $
$$
f(1) = 2(1) + 1 = 2 + 1 = 3
$$

✔ So:
- $ f(-2) = -4 $
- $ f(6) = 13 $
- $ f(1) = 3 $

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#### Graphing Instructions (for Problem 2):

1. For $ x < 1 $: $ y = \frac{x}{2} - 3 $
- Slope = $ \frac{1}{2} $, y-intercept = -3.
- Only defined for $ x < 1 $, so open circle at $ x = 1 $.

Points:
- $ x = 0 $: $ y = 0/2 - 3 = -3 $ → $ (0, -3) $
- $ x = -2 $: $ y = -1 - 3 = -4 $ → $ (-2, -4) $
- $ x = 1 $: $ y = 0.5 - 3 = -2.5 $, but not included, so open circle at $ (1, -2.5) $

2. For $ x \geq 1 $: $ y = 2x + 1 $
- Slope = 2, y-intercept = 1.
- Include $ x = 1 $, so closed circle at $ (1, 3) $

Points:
- $ x = 1 $: $ y = 3 $ → $ (1, 3) $
- $ x = 2 $: $ y = 5 $
- $ x = 3 $: $ y = 7 $
- $ x = 6 $: $ y = 13 $

Draw a ray starting at $ (1, 3) $ with closed circle and go right.

✔ Note: There is a jump discontinuity at $ x = 1 $: left side approaches $ y = -2.5 $, but actual value is $ y = 3 $.

✔ Is this a function? Yes — each input maps to one output.

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✔ Final Answers:

#### Problem 1:
- $ f(3) = -7 $
- $ f(-4) = 1 $
- $ f(-2) = 3 $

#### Problem 2:
- $ f(-2) = -4 $
- $ f(6) = 13 $
- $ f(1) = 3 $

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🔍 Summary:

| Function | Input | Output |
|--------|-------|--------|
| 1 | $ f(3) $ | $ -7 $ |
| 1 | $ f(-4) $ | $ 1 $ |
| 1 | $ f(-2) $ | $ 3 $ |
| 2 | $ f(-2) $ | $ -4 $ |
| 2 | $ f(6) $ | $ 13 $ |
| 2 | $ f(1) $ | $ 3 $ |

Both graphs are functions because they pass the vertical line test.

> 💡 Tip: When graphing piecewise functions, always:
> - Use open circles where the condition does not include equality.
> - Use closed circles where it does.
> - Check continuity at transition points (though not required here).

Let me know if you'd like me to sketch the graphs or provide coordinates for plotting!