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All courses Algebra I · F-IF.1 19 of 76
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 the relation \({(1,~3),~(2,~5),~(3,~5)}\), determine whether it is a function and cite supporting input-output evidence.
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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

Decode the relation

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

In each ordered pair, the first coordinate is an input and the second coordinate is that input's output.

02

Apply the function criterion

\[a~\text{relation}~\text{is}~a~\text{function}~\text{when}~\text{each}~\text{input}~\text{has}~\text{exactly}~\text{one}~\text{output}\]

Repeated output values are allowed; only one input paired with different outputs would violate the definition.

03

Check every source input

\[\text{input}~1~->~3;~\text{input}~2~->~5;~\text{input}~3~->~5\]

Inputs 1, 2, and 3 each occur once and therefore each has exactly one assigned output. Sharing output 5 causes no conflict.

04

Give the evidence-based verdict

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

The relation satisfies the definition, and the exact conclusion cites all three input-output mappings requested by the prompt.

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Another way

  1. Scan only the first coordinates. Since 1, 2, and 3 do not repeat, no input-output conflict is possible.

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Common mistake

Inputs 2 and 3 may share output 5. A function need not be one-to-one; it only forbids a single input from having multiple outputs.

Solution walkthrough video