What are all the possible outputs a function will produce?

In math we have our own vocabulary we use to ask these same questions. The set of inputs a function will accept is called the domain of the function. The set of possible outputs is called the range.

Exercise2.1.1

Consider shopping online at a t-shirt company. Each t-shirt costs \(\$ 13.00\text{,}\) and each time you click on the “add to shopping cart” button, your account adds a charge for \(\$ 13.00\text{.}\)

Let \(C\) represent the cost in dollars of purchasing \(N\) t-shirts. Therefore, \(C = f(N)\text{.}\)

What are the possible input values in this situation? In other words: what is the domain of the function \(f\text{?}\)

What are the possible output values in this situation? In other words: what is the range of the function \(f\text{?}\)

Sometimes a function will only accept certain input values, or it may produce only certain kinds of outputs.

Consider the function:
\begin{equation*}
f(x) = \sqrt{x}
\end{equation*}

This function will take real number inputs, as long as the input is not negative (you can't take the square root of a negative number).

On the other hand, the square root function only produces outputs that are positive or zero.

By graphing the function, the domain and range become more apparent.

The figure shows the graph of \(f\) exists horizontally only from the origin and to the right. Therefore the domain of \(f\) is: \(x \geq 0\text{.}\)

The figure also shows the graph of \(f\) exists vertically from the origin and up. Therefore the range of \(f\) is: \(y \geq 0\text{.}\)

Note: Since the vertical axis is called \(y\text{,}\) it means that \(y = f(x)\text{.}\) Think of \(f(x)\) as the “machine” that turns \(x\) values into \(y\) values. Therefore we can use \(f(x)\) or \(y\) when referring to the range.

Important facts when determining the domain of a function:

You can't take the square root of negative numbers

You can't allow the denominator of a fraction to be zero.

Example2.1.6The algebraic method of finding the domain of any function with a square root

Because we cannot take the square root of a negative number, we can set the expression under the root (called the “radicand”) greater than or equal to zero, and then isolate the variable. The result is the domain of the function.

Let \(f(x)=\sqrt{4x-9}\text{.}\) To find the domain, we set \(4x-9 \geq 0\) and isolate \(x\text{.}\)

The domain is the set of numbers which are greater than or equal to \(\frac{9}{4}\text{,}\) as shown in the graph. The \(x\)-intercept of the graph is the point \(\left(\frac{9}{4}, 0\right)\text{,}\) and the graph extends to the right.

This technique is good only for determining the domain of a function. Finding the range requires other methods.

Figure2.1.7Graph of \(f(x)=\sqrt{4x-9}\)

For functions written as fractions, we must also avoid input values which put the number \(0\) in the denominator. A good way to avoid these numbers is to solve for them: Set the denominator equal to zero, and then solve that equation.

Try some algebraic methods of finding domain on your own. If you get stuck, use your calculator to make a graph of the function. Look to see where the graph exists.

Exercise2.1.10Practice with Roots and Denominators

Identifying the domain and range of a function just by looking at its formula can be difficult. Knowing what to look for comes with experience.

Often, a good way to determine the domain and range of a function is to graph it. Make a graph on a graphing tool and try to identify where the function exists horizontally: this is the domain. Then try to identify where the function exists vertically: this is the range.

Be careful: the domain and range are not necessarily connected intervals. They may consist of different sections that together make up the entire domain or the entire range.