Lesson Explainer: Domain And Range Of A Piecewise Function

In this explainer, we will learn how to find the domain and range of a piecewise-defined function.

We start by recalling what is meant by the domain and range of a function.

Definition: Domain and Range of a Function

The domain of a function is the set of all input values of the function.

The range of a function is the set of all possible outputs of the function, given its domain.

The domain tells us all of the inputs “allowed” for the function. For example, since we cannot input 𝑥=0 into the function 𝑓(𝑥)=1𝑥, as it would be undefined, its domain will not include this value of 𝑥. We can input any other value of 𝑥, so the domain of this function is ℝ−{0}.

The range of a function tells us all of the possible outputs of this function, given its domain. For example, consider the function 𝑓(𝑥)=𝑥, which has a domain of ℝ. Since the square of any real number is nonnegative, 𝑥≥0, therefore, this function only outputs nonnegative real numbers, but we need to check which of these nonnegative real numbers are outputs of the function. To do this, we will show that any nonnegative number is an output of this function. If 𝑦≥0, then 𝑓√𝑦=√𝑦=𝑦.

Hence, the range of this function on ℝ is the set of nonnegative real numbers, given by [0,∞[.

We can use different algebraic techniques and the properties of the function to determine its domain and range. However, it is often easier to do this by using a sketch. Consider the following sketch of 𝑦=𝑓(𝑥).

In the sketch of any function, a point on the curve has the form (𝑥,𝑓(𝑥)), where 𝑥 is the input of the function and 𝑓(𝑥) is the output. In other words, the 𝑥-coordinate of every point on the curve tells us an input of the function and the 𝑦-coordinate tells us an output of the function.

Therefore, we can use the graph of a function to determine its domain and range. To determine the domain of this function, we want to find the 𝑥-coordinate of every point on the curve. We can do this by considering which vertical lines intersect the curve.

For example, if we sketch the vertical line 𝑥=2, we see this intersects our curve at the point (2,3). Hence, 2 is in the domain of our function and 3 is in its range. To determine the full domain of our function, we need to do this with every possible vertical line. We can see any vertical line, 𝑥=𝑐, will intersect this curve. In particular, for 𝑥=0, we have the following:

Since the graph of 𝑦=𝑓(𝑥) has a solid dot at (0,1), we know the function is defined at this point. So, the vertical line 𝑥=0 intersects the curve and 𝑓(0)=1. Therefore, since all vertical lines intersect the curve, its domain is ℝ.

We can find the range of this function by considering horizontal lines.

For example, the line 𝑦=1 intersects the curve at the point (0,1), so 1 is in the range of this function. We can also see that the line 𝑦=0 does not intersect the curve.

Since the curve has a hollow dot at the origin, it does not intersect this horizontal line; in fact, for any 𝑐∈[0,1[, the line 𝑦=𝑐 will not intersect our curve and all other horizontal lines do intersect our curve. Hence, the range of this function is ]−∞,0[∪[1,∞[.

Before we discuss finding the domain and range of a piecewise-defined function, let’s start by recalling what we mean by these types of functions.

Definition: Piecewise Function

A piecewise function is a function that consists of multiple subfunctions, with each subfunction defined over a subset of the main function’s domain, called a subdomain.

The equation of a piecewise function is written with a curly bracket to indicate that it is comprised of more than one subfunction. An example of a piecewise function is 𝑓(𝑥)=𝑥,𝑥