How to Determine Whether the Relation is a Function

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In mathematics, a function is a rule that relates every element in one set, called the domain, to exactly one element in another set, called the range. On an ​x​-​y​ axis, the domain is represented on the ​x​-axis (horizontal axis) and the domain on the ​y​-axis (vertical axis). A rule that relates one element in the domain to more than one element in the range is not a function. This requirement means that, if you graph a function, you cannot find a vertical line that crosses the graph in more than one place.

TL;DR (Too Long; Didn't Read)

A relation is a function only if it relates each element in its domain to only one element in the range. When you graph a function, a vertical line will intersect it at only one point.

Mathematical Representation 

Mathematicians usually represent functions by the letters "​f​(​x​)," although any other letters work just as well. You read the letters as "​f​ of ​x​." If you choose to represent the function as ​g​(​y​), you would read it as "​g​ of ​y​." The equation for the function defines the rule by which the input value ​x​ is transformed into another number. There are an infinite number of ways to do this. Here are three examples:

f(x) = 2x \\ \,\\ g(y) = y^2 + 2y + 1 \\ \,\\ p(m) = \frac{1}{\sqrt{m - 3}}

Determining the Domain

The set of numbers for which the function "works" is the domain. This can be all numbers, or it can be a specific set of numbers. The domain can also be all numbers except one or two for which the function doesn't work. For example, the domain for the function

f(x) = \frac{1}{2-x}

is all numbers except 2, because when you input two, the denominator is 0, and the result is undefined. The domain for

\frac{1}{4 - x^2}

on the other hand, is all numbers except +2 and −2 because the square of both of these numbers is 4.

You can also identify the domain of a function by looking at its graph. Starting at the extreme left and moving to the right, draw vertical lines through the ​x​-axis. The domain is all the values of ​x​ for which the line intersects the graph.

When Is a Relation Not a Function?

By definition, a function relates each element in the domain to only one element in the range. This means that each vertical line you draw through the ​x​-axis can intersect the function at only one point. This works for all linear equations and higher-power equations in which only the x term is raised to an exponent. It doesn't always work for equations in which both the ​x​ and ​y​ terms are raised to a power. For example, ​x2 + ​y2 = ​a2 defines a circle. A vertical line can intersect a circle at more than one point, so this equation is not a function.

In general, a relationship ​f​(​x​) = ​y​ is a function only if, for each value of ​x​ that you plug into it, you get only one value for ​y​. Sometimes the only way to tell if a given relationship is a function or not is to try various values for x to see if they yield unique values for ​y​.

Examples: ​Do the following equations define functions?

y = 2x +1

This is the equation of a straight line with slope 2 and ​y​-intercept 1, so it ​IS​ a function.

y^2 = x + 1

Let ​x​ = 3. The value for y can then be ±2, so this ​IS NOT​ a function.

y^3 = x^2

No matter what value we set for ​x​, we'll get only one value for ​y​, so this ​IS​ a function.

y^2 = x^2

Because ​y​ = ±√​x2, this ​IS NOT​ a function.

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