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All courses Math I · F-IF.1 19 of 59
Understand functions as mappings from inputs to exactly one output; connect f(x) to graph y=f(x).

Decide whether an ordered-pair set represents a function

Problem
For relation (1,3),(2,5),(3,5), identify function verdict and input-output evidence. If it is a function, list every input with its single output; otherwise identify one input with two outputs.
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Hint

Check whether any single input is paired with more than one output.

A relation is a function when every input has exactly one output. Repeated outputs are allowed; repeated inputs with different outputs are not.

Solution walkthrough

01

Read the relation

\[(1,3),~(2,5),~(3,5)\]

Start by looking at the listed inputs and outputs or the mapping description.

02

Check the inputs

\[\text{inputs}~1,~2,~\text{and}~3~\text{each}~\text{appear}~\text{once}\]

Look for any input value that is paired with more than one output value.

03

Apply the function definition

\[\text{No}~\text{input}~\text{is}~\text{repeated}~\text{with}~\text{two}~\text{different}~\text{outputs},~\text{so}~\text{the}~\text{relation}~\text{is}~a~\text{function}.\]

Use the result of the input check to decide whether the relation is a function.

04

State the conclusion

\[\text{verdict}~=~\text{function};~\text{evidence}~=~1→3,~2→5,~3→5,~\text{so}~\text{every}~\text{input}~\text{has}~\text{exactly}~\text{one}~\text{output}\]

The final answer should say yes or no and explain why.

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Another way

  1. The quickest check is to scan the inputs only. Outputs may repeat without breaking the function rule.

!

Common mistake

A common mistake is to think repeated outputs mean the relation is not a function. The real issue is whether one input has more than one output.