Problem Explanation:
The task is to determine whether each given relation (represented as a mapping diagram) is a
function. A relation is considered a function if
each element in the domain (input set) maps to exactly one element in the range (output set). In other words, no input can be associated with more than one output.
Solution Approach:
1.
Identify the Domain and Range:
- The left oval represents the domain (inputs).
- The right oval represents the range (outputs).
2.
Check for Functionality:
- For each input in the domain, verify that it maps to
exactly one output in the range.
- If any input maps to
more than one output, the relation is
not a function.
- If every input maps to
exactly one output, the relation is a
function.
3.
Analyze Each Diagram:
Let's analyze each of the six diagrams step by step.
---
Diagram 1:
-
Domain: \(-1, 0, 4, 7, 8, 12, 13\)
-
Range: \(-6, -2, 5, 6, 8, 17\)
-
Mapping:
- \(-1 \to -6\)
- \(0 \to -2\)
- \(4 \to 5\)
- \(7 \to 6\)
- \(8 \to 8\)
- \(12 \to 17\)
- \(13 \to 17\)
-
Analysis: Each input maps to exactly one output.
-
Conclusion: This is a function.
---
Diagram 2:
-
Domain: \(-19, -14, -11, -9, 2, 4\)
-
Range: \(-16, -13, -11, -9, 9, 10\)
-
Mapping:
- \(-19 \to -16\)
- \(-14 \to -13\)
- \(-11 \to -11\)
- \(-9 \to -9\)
- \(2 \to 9\)
- \(4 \to 10\)
-
Analysis: Each input maps to exactly one output.
-
Conclusion: This is a function.
---
Diagram 3:
-
Domain: \(-5, -2, 0, 3, 6, 7\)
-
Range: \(10, 11, 14, 15, 17, 19\)
-
Mapping:
- \(-5 \to 10\)
- \(-2 \to 11\)
- \(0 \to 14\)
- \(3 \to 15\)
- \(6 \to 17\)
- \(7 \to 19\)
-
Analysis: Each input maps to exactly one output.
-
Conclusion: This is a function.
---
Diagram 4:
-
Domain: \(4, 5, 7, 10, 11, 12, 15\)
-
Range: \(1, 3, 4, 8, 9\)
-
Mapping:
- \(4 \to 1\)
- \(5 \to 3\)
- \(7 \to 4\)
- \(10 \to 8\)
- \(11 \to 9\)
- \(12 \to 8\)
- \(15 \to 9\)
-
Analysis: Each input maps to exactly one output.
-
Conclusion: This is a function.
---
Diagram 5:
-
Domain: \(-19, -18, -16, -14, -12, -10, -8\)
-
Range: \(-20, -14, -12, -11, -10, -7, -4\)
-
Mapping:
- \(-19 \to -20\)
- \(-18 \to -14\)
- \(-16 \to -12\)
- \(-14 \to -11\)
- \(-12 \to -10\)
- \(-10 \to -7\)
- \(-8 \to -4\)
-
Analysis: Each input maps to exactly one output.
-
Conclusion: This is a function.
---
Diagram 6:
-
Domain: \(-7, -5, 0, 1, 5, 10, 14\)
-
Range: \(-11, 1, 4, 5, 17\)
-
Mapping:
- \(-7 \to -11\)
- \(-5 \to 1\)
- \(0 \to 4\)
- \(1 \to 5\)
- \(5 \to 17\)
- \(10 \to 5\)
- \(14 \to 17\)
-
Analysis: Each input maps to exactly one output.
-
Conclusion: This is a function.
---
Final Answers:
1.
Yes
2.
Yes
3.
Yes
4.
Yes
5.
Yes
6.
Yes
Boxed Answer:
\[
\boxed{\text{Yes, Yes, Yes, Yes, Yes, Yes}}
\]
Parent Tip: Review the logic above to help your child master the concept of identifying functions worksheet pdf.
