Precalculus: An Active Reading Approach: Elementary Functions

Subsection 2.3.1 Domain and Range

Sometimes a formula can only accept certain kinds of input values. For instance the square root function

can only accept values of \(x\) that are not negative, because only non-negative inputs allow the function to yield real-valued outputs.

In math we say

The function \(f\) has a domain such that \(x \geqslant 0\)

meaning whatever values we choose for \(x\text{,}\) they must be greater than or equal to zero. That's the math way of saying “not negative”.

The domain of a function is the set of all inputs a function is allowed to use.

Once we find the domain of a function, we can then consider all the possible outputs for the function. The set of all possible outputs is called the range of the function.

At this point it is important to note that when we state the domain of a function, we refer to the input variable. So far our inputs have been called \(x\text{,}\) so every domain we have described has had \(x\) in it. For instance, \(f(x) = \frac{x}{{{x^2} - 1}}\) had the domain \(\{x \vert x \neq \pm 1 \}\text{.}\)

When we state the range of a function, we must refer to the output. If \(y=f(x)\) where \(f(x) = \frac{x}{{{x^2} - 1}}\text{,}\) then our range has to refer to \(y\) or to \(f(x)\text{,}\) both of which represent the output. In this case the range could be written \(y \geqslant 0\) or \(f(x) \geqslant 0\text{.}\)

Identifying the domain and range directly from a formula can be difficult. It is often more clear if we look at functions graphically. Below is a graph of \(f(x) = \sqrt {x - 8}\)

Notice the graph only appears for input values \(x \geqslant 8\text{.}\) That is the domain of \(f\text{.}\)

Also notice the graph only appears at or above the \(x-\)axis. The graph shows us the range is \(y \geqslant 0\text{.}\)

When you want to find the domain and range of a function, one option is to graph it on a graphing tool. Determine the intervals or boundaries on the horizontal axis where the graph exists. This is the domain.

Next, use the graph to determine if there are vertical intervals or boundaries to how high or low the graph reaches. This is the range.

Find the domain and range of the function \(f(x)\) graphed below.

Solution

The graph only exists when \(x\) is greater than \(-1\) and less than or equal to \(5\text{.}\) This is the domain, and it can be written as an inequality

\begin{equation*} -1 \lt x \leq 5 \end{equation*}

or an interval

\begin{equation*} (-1,5]\text{.} \end{equation*}

The graph curves up and down, but it reaches every output value where \(y\) is greater than or equal to \(-27\) and less than or equal to \(25\text{.}\) This is the range, and it can be written as an inequality

\begin{equation*} -27 \leq y \leq 25 \end{equation*}

or an interval

\begin{equation*} [-27,25]\text{.} \end{equation*}

In the next exercise, you will manipulate a graph to change its domain and range.

Subsection 2.3.2 Piecewise Defined Functions

A piecewise defined function is actually a collection of two or more functions that tell us how to use an input to get an output. Each function must state the domain on which it is defined, that way you know when to use it.

We start with some easy, everyday examples of piecewise defined functions.

The cost of renting a snowboard is \(\$25\) per day, or \(\$60\) for three days, or \(\$75\) for a week.

The first cost function is \(\$25\) per day and is good for \(1\) or \(2\) days. The domain of this function is the “\(1\) or \(2\) days”. It tells you when to use this cost function.

On the third day is a new cost function, a simple constant of \(\$60\text{.}\)

Finally, the constant function \(\$75\) is used when we keep the snowboard for more than \(3\) days, up to \(7\) days (a full week).

Each rule has its own domain that tells us when to use the rule. In Figure 2.3.10, we see the graph of the price function. Notice the graph has a constant output of \(50\) when \(1 \lt x \leq 2\text{,}\) but it immediately changes to \(60\) when \(x \gt 2\text{.}\)

To create a piecewise function, start by finding formulas or rules for all the different functions in the set. Then identify the domain for each function.

When you finally write the piecewise function, write each formula with its corresponding domain to its right.

Subsection 2.3.3 Conclusion

Test your understanding of this chapter with the following exercises.

Student Learning Outcome: Determine the domain and range of a function.

Student Learning Outcome: Define, distinguish, and apply piecewise-defined functions.

Student Learning Outcome: Determine from a graph or an equation whether a function is continuous or discontinuous.