Functions, Domain, and Range Study Guide (page 2)
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.
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.
Domain and Range
Earlier, we looked at the equation y =√x and stated that x could not be negative, because we cannot take the square root of a negative number. However, for the equation y = x, we could make x any real number. For this equation, we say that the domain is all real numbers. The domain of an equation or function is all the values that can be substituted for x. In the equation y =√x, the domain is zero and all positive real numbers.
While the domain describes what kind of x values can be put into an equation, the range tells us what kind of y values we will get back. In the equation y = x, because x can be any real number, y can be any real number. The range of y = x is all real numbers. The equation y =√x has a domain of zero and all positive real numbers, and the possible values of y are also zero and all positive real numbers. So the domain and range of this function are the same as well.
The domain and range are not always the same. The equation y = x2 has a domain of all real numbers, because any real number can be substituted for x. However, the square of any real number, including negative real numbers, is always positive. There will be no y values that are negative, so the range of y = x2 is zero and all positive real numbers.
What are the domain and range of ? The denominator of a fraction can never be 0, which means that x cannot be zero. The domain of the equation is all real numbers except 0. When these values are put into the equation, we can get back any y value except 0. The range of is all real numbers except 0.
The value that makes a fraction undefined is often not only a value that must be excluded from the domain, but also a value that must be excluded from the range
Find practice problems and solutions for these concepts at Functions, Domain, and Range Practice Questions.
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