How to tell if a relation is a function with the vertical line test

When we look at a graph, a table, or a list of points, we can ask the same question. Does each input have just one output? Let’s see how to check that in each form.

Look for an input with two different outputs

A function assigns exactly one output to each input in its domain. Two different inputs can share the same output. For example, the points $(1, 4)$ and $(2, 4)$ can both belong to a function because each input still has just one output.

The set $(1, 2)$ , $(2, 4)$ , $(2, 3)$ fails because the input $2$ has two outputs, $4$ and $3$ . The set $(1, - 3)$ , $(2, - 4)$ , $(3, - 5)$ passes because none of its inputs has more than one output.

Use the vertical line test on a graph

A vertical line keeps $x$ fixed. If it meets the graph twice, the same input has two different $y$ -values, so the graph fails the test. A function’s graph meets every vertical line at most once. A line at an $x$ -value outside the domain misses the graph entirely.

The dashed line marks one input. Each dot on it is a different output for that input.

An upright parabola such as $y = x^{2}$ passes the test because each $x$ gives one value when we square it. A circle fails because some vertical lines cross it twice.

Check a relation directly from its equation

For $x^{2} + y^{2} = 1$ , try $x = 0$ . The equation becomes $y^{2} = 1$ , so $y$ can be $1$ or $- 1$ . This is the same test in algebra form. One input has two outputs, so the circle does not give $y$ as a function of $x$ .

Learn mathKrista KingNovember 7, 2020math, learn online, online course, online math, algebra, algebra 1, algebra i, functions, testing for functions, function test, vertical line test, vertical line, functions as point sets, functions as coordinate points

How to tell if a relation is a function with the vertical line test

Look for an input with two different outputs

Use the vertical line test on a graph

Check a relation directly from its equation

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