What Makes a Relation a Function
F.IF.1
Read the standard text
F.IF.1 Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).
Section 1 · LearnLearn the idea
In this lesson you will: A relation pairs inputs with outputs; it is a function when each input is paired with exactly one output; Repeated outputs are allowed. Repeated inputs with different outputs are not; On a graph, the test is vertical: one input, one point above it.
One input, one output — and that is the whole definition
A relation is any pairing of inputs with outputs: a list of ordered pairs, a table, an arrow diagram, a graph. A function is a relation with one extra promise. Hand it an input, and it hands back exactly one output. Every time. No choices, no ties.
The set of allowed inputs is the domain. The set of outputs the rule actually produces is the range. Lesson 7.4 works those two properly; for now, notice that the definition of a function only makes a demand about the domain side.
One bad input spoils the whole thing
A relation fails to be a function the moment a single input appears twice with different partners. It does not matter how well behaved the other ninety-nine inputs are.
The reverse situation is fine, and this is where most of the confusion lives. Three different inputs may all send you to 6. A function is allowed to be forgetful about where you came from; it is not allowed to be undecided about where you are going.
The vertical line test, and what it is actually testing
A point on a graph is an input-output pair: the horizontal position is the input, the height is the output. So \"input 3 has two different outputs\" shows up as two points stacked above x=3.
That is all the vertical line test is. Sweep an imaginary vertical line across the graph. If it ever touches the curve in two places at once, that input has two outputs and the graph is not a function of x. The test is not a separate rule to remember — it is the definition, drawn.
Common mistake — Deciding it is not a function because an output repeats
Shown \(1,6),(2,6),(3,6)\, a student writes \"not a function — 6 appears three times.\"
The instinct is reasonable. The word that gets remembered is \"one to one,\" and a repeated number does look like a collision.
But the definition only restricts the input side. Three inputs sharing an output is a perfectly ordinary function — every student in a class can have the same lunch period. What is forbidden is one input pointing at two outputs, because then the rule has no answer to give.
Drawn as a graph, \(1,6),(2,6),(3,6)\ is three dots in a row at the same height. No vertical line hits two of them.
How to catch it: Cover the output column with your hand and look only at the inputs. If no input is listed twice, it is a function — whatever the outputs are doing.
Section 2 · DoPractice this standard
These published practice sets are scored on the student site and cover F.IF.1 directly.