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# Functions, Domain, and Range Study Guide

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Updated on Oct 3, 2011

## 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.

#### Tip:

When you are trying to decide whether an equation is a function, always ask: Is there a value of x having two y values? If so, the equation is not a function.

## 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.

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