Let's solve and complete the worksheet titled
"Relations & Functions Notes" step by step. The goal is to fill in the blanks based on mathematical definitions and examples provided.
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🔹 Definitions at the Top
relation - A
relationship between sets of values. The two types of values are called
inputs and
outputs.
>
✔ Explanation: A *relation* is a set of ordered pairs that shows how elements from one set (input) are related to elements in another set (output).
function - A special type of
relation where each input has only
one output.
>
✔ Explanation: In a function, every input (x-value) must correspond to exactly one output (y-value). This is known as the
vertical line test for graphs.
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🔹 Ways to Represent Relations
#### 1.
Ordered Pairs
-
Definition: The
x coordinate and
y coordinate of a point.
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✔ Explanation: Each ordered pair $(x, y)$ represents an input $x$ and its corresponding output $y$.
#### 2.
X-Y Table
-
Definition: A table that shows inputs and outputs in terms of
x and
y.
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✔ Explanation: The left column lists the inputs ($x$), and the right column lists the outputs ($y$).
#### 3.
Mapping Diagram
-
Definition: A type of diagram that shows the relationship between each
input and
output by using arrows.
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✔ Explanation: Arrows go from each input value to its corresponding output(s). If any input points to more than one output, it’s
not a function.
#### 4.
Graph
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Definition: A visual display of
points.
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✔ Explanation: Plots the ordered pairs $(x, y)$ on the coordinate plane. A graph represents a relation visually.
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✔ Final Completed Worksheet:
| WORD | DEFINITION | EXAMPLE |
|------|-----------|--------|
| ORDERED PAIRS | The
x coordinate and
y coordinate of a point. | $(x, y)$<br>$(1,3), (2,1), (3,4), (5,2)$ |
| X-Y TABLE | A table that shows inputs and outputs in terms of
x and
y. |
| MAPPING DIAGRAM | A type of diagram that shows the relationship between each
input and
output by using arrows. |
| GRAPH | A visual display of
points. |
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📌 Additional Notes:
- All four representations show the same relation:
$\{(1,3), (2,1), (3,4), (5,2)\}$
- Is this a
function?
✔ Yes — because each input (1, 2, 3, 5) maps to
only one output.
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✔ Summary of Answers:
1.
relation – A
relationship between sets of values. The two types of values are called
inputs and
outputs.
2.
function – A special type of
relation where each input has only
one output.
3.
Ordered Pairs – The
x coordinate and
y coordinate of a point.
4.
X-Y Table – Shows inputs and outputs in terms of
x and
y.
5.
Mapping Diagram – Shows relationship between each
input and
output using arrows.
6.
Graph – A visual display of
points.
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