Functions, Domain, and Range Study Guide

Introduction to Functions, Domain, and Range

Mathematics is an escape from reality

—Stanislaw Ulam (1909–1984) Polish Mathematician

In this lesson, you'll learn how to determine if an equation is a function, and how to find the domain and range of a function.

Almost every line we have seen in the last few lessons has been a function. An equation is a function if every x value has no more than one y value. For instance, the equation y = 2x is a function, because there is no value of x that could result in two different y values. When an equation is a function, we can replace y with f(x), which is read as"f of x." If you see an equation written as f(x) = 2x, you are being told that y is a function of x, and that the equation is a function.

The equation x = 5 is not a function. When x is 5, y has many different values. Vertical lines are not functions. They are the only type of line that is not a function.

What about the equation y = x² Positive and negative values of x result in the same y value, but that is just fine. A function can have y values that each have more than one x value, but a function cannot have x values that each have more than one y value. There is no number that can be substituted for x that results in two different y values, so y = x² is a function

What about the equation y² = x? In this case, two different y values, such as 2 and –2, result in the same x value, 4, so y² = x is not a function. We must always be careful with equations that have even exponents. If we take the square root of both sides of the equation y² = x, we get y = √x, which is a function. x cannot be negative, because we cannot find the square root of a negative number. Because x must be positive, y must be positive. This means that, unlike y² = x, there is no x value having two y values.

Vertical Line Test

When we can see the graph of an equation, we can easily identify whether the equation is a function by using the vertical line test. If a vertical line can be drawn anywhere through the graph of an equation, such that the line crosses the graph more than once, then the equation is not a function. Why? Because a vertical line represents a single x value, and if a vertical line crosses a graph more than once, then there is more than one y value for that x value.

Look at the following graph. We do not know what equation is shown, but we know that it is a function, because there is no place on the graph where a vertical line will cross the graph more than once.

Even if there are many places on a graph that pass the vertical line test, if there is even one point for which the vertical line test fails, then the equation is not a function. The following graph, a circle, is not a function, because there are many x values that have two y values. The dark line drawn where x = 5 shows that the graph fails the vertical line test. The line crosses the circle in two places.