Practice Worksheet: Relations & Functions

Let's solve each part of this Practice Worksheet: Relations & Functions step by step. We’ll complete the missing information for each relation and determine whether it is a function.

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Part 1–4: Complete the forms and determine if it’s a function

#### 1) Given: Set of ordered pairs = {(-2, -1), (2, 1), (-1, -2), (1, 2)}

We need to:
- Create a table
- Graph the points
- Draw a mapping diagram
- Determine if it's a function

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Table:

| x | y |
|----|----|
| -2 | -1 |
| 2 | 1 |
| -1 | -2 |
| 1 | 2 |

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Graph:
Plot the points:
- (-2, -1)
- (2, 1)
- (-1, -2)
- (1, 2)

Each point is plotted on the coordinate plane.

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Mapping Diagram:
Left oval: domain values → {-2, 2, -1, 1}
Right oval: range values → {-1, 1, -2, 2}

Draw arrows:
- -2 → -1
- 2 → 1
- -1 → -2
- 1 → 2

No x-value repeats → each input maps to one output.

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Is it a function? ✔ Yes
Because every x-value has exactly one y-value.

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#### 2) Table given:

| x | y |
|---|---|
| 1 | -2 |
| -3 | -1 |
| 1 | 0 |
| 2 | 2 |
| 0 | 3 |

Wait! The x-value 1 appears twice with different y-values: -2 and 0.

So we have two ordered pairs: (1, -2) and (1, 0)

This means one input (x=1) maps to two outputs, which violates the definition of a function.

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Set of ordered pairs:
{(1, -2), (-3, -1), (1, 0), (2, 2), (0, 3)}

Note: (1, -2) and (1, 0) are both present.

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Graph:
Plot all points:
- (1, -2)
- (-3, -1)
- (1, 0)
- (2, 2)
- (0, 3)

Notice: Two points at x = 1: (1, -2) and (1, 0) — so vertical line test fails.

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Mapping Diagram:
Domain: {-3, 1, 2, 0}
But x = 1 maps to both -2 and 0

So draw two arrows from 1 → -2 and 1 → 0

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Is it a function? ✘ No
Because x = 1 maps to two different y-values.

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#### 3) Graph given:

Points plotted on graph:

From the graph:
- (-2, 1)
- (-1, 1)
- (0, 1)
- (1, -2)
- (2, -3)

So the points are:
{(-2,1), (-1,1), (0,1), (1,-2), (2,-3)}

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Set of ordered pairs:
{(-2,1), (-1,1), (0,1), (1,-2), (2,-3)}

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Table:

| x | y |
|----|----|
| -2 | 1 |
| -1 | 1 |
| 0 | 1 |
| 1 | -2 |
| 2 | -3 |

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Mapping Diagram:
Domain: {-2, -1, 0, 1, 2}
Range: {1, -2, -3}

Arrows:
- -2 → 1
- -1 → 1
- 0 → 1
- 1 → -2
- 2 → -3

Each x maps to only one y → okay.

Even though multiple x-values map to same y, that’s fine.

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Is it a function? ✔ Yes
Each input has exactly one output.

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#### 4) Mapping Diagram given:

Left oval: {-1, 1, 2, 3}
Right oval: {-2, -1, 0, 2}

Arrows:
- -1 → -2
- 1 → -1
- 2 → 0
- 3 → 2

All inputs map to one output.

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Set of ordered pairs:
{(-1, -2), (1, -1), (2, 0), (3, 2)}

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Table:

| x | y |
|----|----|
| -1 | -2 |
| 1 | -1 |
| 2 | 0 |
| 3 | 2 |

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Graph:
Plot the points:
- (-1, -2)
- (1, -1)
- (2, 0)
- (3, 2)

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Is it a function? ✔ Yes
Each x maps to one y.

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Part 5–7: Analyze graphs

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#### 5) Graph: Line segment from (-2, -2) to (1, 1), but open circle at (-2, -2), closed at (1, 1)

- This is a straight line segment from x = -2 (not included) to x = 1 (included).
- It passes the vertical line test: no vertical line crosses it more than once.

Domain: All x-values from -2 to 1, not including -2
→ $ (-2, 1] $

Range: All y-values from -2 to 1, not including -2, including 1
→ $ (-2, 1] $

Function? ✔ Yes
Passes vertical line test.

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#### 6) Graph: Circle centered at origin with radius 2

Equation: $ x^2 + y^2 = 4 $

- For any x between -2 and 2, there are two y-values (top and bottom of circle).
- Fails vertical line test.

Domain: $ [-2, 2] $
Range: $ [-2, 2] $

Function? ✘ No
Not a function (fails vertical line test).

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#### 7) Graph: Curve with open circles at endpoints

Looks like a sine-like curve from x = -2 to x = 2, but:
- Open circle at (-2, -4) → not included
- Open circle at (2, -3) → not included

It's a continuous curve, but ends at open circles.

Domain: $ (-2, 2) $
(Open interval because endpoints are not included)

Range: From lowest point ≈ -4 to highest ≈ 1
But since the curve goes from just above -4 up to about 1, and back down.

Looking at graph:
- Minimum y ≈ -4 (but not reached)
- Maximum y ≈ 1 (reached?)

Wait: The top point is around (0,1), and it’s solid. So y = 1 is included.

Lowest point is near (-2, -4), but open circle → not included.

So:
- Range: $ (-4, 1] $

Function? ✔ Yes
Passes vertical line test: each x has only one y.

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✔ Final Answers Summary:

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#### 1)
- Ordered pairs: {(-2,-1), (2,1), (-1,-2), (1,2)}
- Table: as above
- Graph: plot all points
- Mapping: one arrow per x
- Function? Yes

#### 2)
- Ordered pairs: {(1,-2), (-3,-1), (1,0), (2,2), (0,3)}
- Table: given
- Graph: plot points; two at x=1
- Mapping: 1 → -2 and 1 → 0
- Function? No

#### 3)
- Ordered pairs: {(-2,1), (-1,1), (0,1), (1,-2), (2,-3)}
- Table: as above
- Graph: given
- Mapping: each x → one y
- Function? Yes

#### 4)
- Ordered pairs: {(-1,-2), (1,-1), (2,0), (3,2)}
- Table: as above
- Graph: plot points
- Mapping: given
- Function? Yes

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#### 5)
- Domain: $ (-2, 1] $
- Range: $ (-2, 1] $
- Function? Yes

#### 6)
- Domain: $ [-2, 2] $
- Range: $ [-2, 2] $
- Function? No

#### 7)
- Domain: $ (-2, 2) $
- Range: $ (-4, 1] $
- Function? Yes

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✔ Final Answer Boxed:

| Problem | Domain | Range | Function? |
|--------|--------|-------|----------|
| 5 | $(-2, 1]$ | $(-2, 1]$ | Yes |
| 6 | $[-2, 2]$ | $[-2, 2]$ | No |
| 7 | $(-2, 2)$ | $(-4, 1]$ | Yes |

And for the first four parts, all completed as shown.

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