Identifying Functions from Graphs, Tables, and Equations

Identifying Functions in Any Form

By the end of this lesson, you will be able to decide whether a relation is a function whether it arrives as pairs, a mapping, a table, a graph or an equation, and say exactly why in each case.

One Rule, Five Disguises

The definition has not changed since the first lesson of this unit:

Every input has exactly one output.

What changes is where the inputs are hiding. A relation can arrive in five different costumes, and the whole skill is finding the inputs quickly and then asking the one question.

Form	Where the inputs are	The question
Ordered pairs	The first coordinates	Does any repeat with a different partner?
Mapping diagram	The left-hand column	Does any left dot have two arrows leaving it?
Table	The $x$ row or column	Does any $x$ appear twice with different $y$?
Graph	Horizontal position	Does a vertical line cross it twice?
Equation	Whatever you solve for $y$ from	Does one $x$ give more than one $y$?

Five rows, one question. Everything below is practice at recognizing which row you are in.

Mapping Diagrams: Count the Arrows Out

A mapping diagram draws inputs on the left, outputs on the right, and an arrow for each pair.

Not a function: a left dot with TWO arrows leaving it. That input has two answers.

Still a function: a right dot with two arrows arriving. Two inputs sharing an output, which has always been allowed.

Arrows out, not arrows in. Said that way it is the same rule as everywhere else, and it is the version that survives being asked at speed.

Sort It: Function or Not

Each description below is a relation in one of the five forms.

Example 1: The Two Lines That Test Different Things

Here is the whole method on one graph, worked all the way through. Take $y = x^2$, an ordinary parabola.

• A vertical line at x = 3 crosses the parabola how many times? 1

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