The image you provided contains several examples related to
relations and functions, which are fundamental concepts in algebra. Let's analyze each part of the image step by step:
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1. Table Representation
The first table shows a relationship between \( x \) and \( y \):
| \( x \) | \( y \) |
|---------|---------|
| 5 | 1 |
| 5 | 2 |
| 5 | 3 |
| 5 | 4 |
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Analysis:
- Here, the value of \( x \) is always 5, but \( y \) takes on multiple values (1, 2, 3, 4).
- This means that for a single input \( x = 5 \), there are multiple outputs (\( y \)).
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Conclusion: This is not a function because a function must have exactly one output for each input.
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2. Mapping Diagram
The mapping diagram shows arrows from inputs to outputs:
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Top Mapping Diagram:
- Inputs: 6, 8, 11, 14
- Outputs: 9, 11, 15, 20
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Bottom Mapping Diagram:
- Inputs: 6, 8, 10, 12, 14
- Outputs: -4, 2, 6, -4, 11
####
Analysis:
- In the
top mapping diagram, each input maps to exactly one output. For example:
- 6 → 9
- 8 → 11
- 11 → 15
- 14 → 20
-
Conclusion: This is a function.
- In the
bottom mapping diagram, each input also maps to exactly one output. For example:
- 6 → -4
- 8 → 2
- 10 → 6
- 12 → -4
- 14 → 11
-
Conclusion: This is also a function.
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3. Graph Representation
The graph shows points plotted on a coordinate plane:
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Points:
- (6, 9)
- (8, 12)
- (10, 6)
- (12, 4)
- (14, 6)
####
Analysis:
- Each \( x \)-value corresponds to exactly one \( y \)-value.
- For example:
- \( x = 6 \) maps to \( y = 9 \)
- \( x = 8 \) maps to \( y = 12 \)
- \( x = 10 \) maps to \( y = 6 \)
- \( x = 12 \) maps to \( y = 4 \)
- \( x = 14 \) maps to \( y = 6 \)
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Conclusion: This is a function because each input has exactly one output.
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4. Definition of Function
A relation is a
function if every element in the domain (input set) is paired with exactly one element in the range (output set). If any input is associated with more than one output, it is not a function.
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Final Answer:
Based on the analysis:
1. The table at the top is
not a function because \( x = 5 \) maps to multiple \( y \)-values.
2. Both mapping diagrams are
functions because each input maps to exactly one output.
3. The graph is also a
function because each \( x \)-value corresponds to exactly one \( y \)-value.
Thus, the key takeaway is:
\[
\boxed{\text{A relation is a function if each input has exactly one output.}}
\]
Parent Tip: Review the logic above to help your child master the concept of functions worksheet algebra 1.
