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Which of the following relations represents a function from π to π?
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And then we have four possible relations given by mapping diagrams.
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Now remember, a mapping diagram takes elements from one set π and maps them onto elements in a second set, in this case, π.
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Then, a function will always be an example of a relation, but we cannot say the reverse is necessarily true.
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All relations will not necessarily be functions.
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And this is because a function must map each element from the input, set π, onto exactly one element of the output, set π.
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So to establish which of the relations represents a function from π to π, we need to identify which of the relations maps one element from set π onto exactly one in set π.
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Letβs begin by considering our first relation.
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This relation maps negative two onto negative four in set π.
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Similarly, it takes the element zero from set π and maps it onto one element in set π.
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Finally, the element one is mapped onto exactly one element in set π, negative four.
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Since each element in set π is mapped onto only one element in set π, relation one must be a function.
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For completeness, letβs double check (R2), (R3), and (R4).
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Letβs first look at (R2).
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If we look carefully, we see that this element negative two in set π has two arrows coming from it.
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It is mapped onto negative four in set π and negative two.
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Since this element negative two maps onto two elements in the output, we know that (R2), the second relation, cannot be a function.
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Similarly, consider element one in relation three.
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This is mapped onto the element negative nine and the element negative two in set π.
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Since this element does not map onto exactly one element in set π, we disregard (R3).
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This relation does not represent a function.
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And finally, in our fourth relation, the element negative two maps onto both the element negative nine and the element negative two in set π.
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It is not one to one.
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This element maps onto two elements of the output, so we disregard relation four.
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And so (R1) relation one must represent a function from π to π.