Problem Explanation:
The task is to determine whether each given relation represents a
function. A relation is considered a function if every element in the domain (input set) is paired with exactly one element in the codomain (output set). In other words, no input can be associated with more than one output.
Solution Approach:
1.
Identify the Domain and Codomain:
- The left oval contains the domain (inputs).
- The right oval contains the codomain (outputs).
- Arrows indicate how elements from the domain are mapped to elements in the codomain.
2.
Check for Functionality:
- For each element in the domain, ensure it is connected to
exactly one element in the codomain.
- If any element in the domain is connected to
more than one element in the codomain, the relation is
not 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\}
-
Codomain: \{-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 element in the domain is mapped to exactly one element in the codomain.
-
Conclusion: This is a function.
---
Diagram 2:
-
Domain: \{-19, -14, -11, -9, 2, 4\}
-
Codomain: \{-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 element in the domain is mapped to exactly one element in the codomain.
-
Conclusion: This is a function.
---
Diagram 3:
-
Domain: \{-5, -2, 0, 3, 6, 7\}
-
Codomain: \{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 element in the domain is mapped to exactly one element in the codomain.
-
Conclusion: This is a function.
---
Diagram 4:
-
Domain: \{4, 5, 7, 10, 11, 12, 15\}
-
Codomain: \{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 element in the domain is mapped to exactly one element in the codomain.
-
Conclusion: This is a function.
---
Diagram 5:
-
Domain: \{-19, -18, -16, -14, -12, -10, -8\}
-
Codomain: \{-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 element in the domain is mapped to exactly one element in the codomain.
-
Conclusion: This is a function.
---
Diagram 6:
-
Domain: \{-7, -5, 0, 1, 5, 10, 14\}
-
Codomain: \{-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 element in the domain is mapped to exactly one element in the codomain.
-
Conclusion: This is a function.
---
Final Answers:
1.
Yes
2.
Yes
3.
Yes
4.
Yes
5.
Yes
6.
Yes
Thus, the final answer is:
\[
\boxed{\text{Yes, Yes, Yes, Yes, Yes, Yes}}
\]
Parent Tip: Review the logic above to help your child master the concept of function or not a function worksheet with answers.
