How To Find The Domain Of Piecewise Functions

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

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

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 the states all of the inputs "allowed" for the office. For instance, since nosotros cannot input into the office , as information technology would be undefined, its domain will not include this value of . We can input any other value of , so the domain of this function is .

The range of a function tells us all of the possible outputs of this office, given its domain. For example, consider the function , which has a domain of . Since the square of whatsoever real number is nonnegative, , therefore, this role simply outputs nonnegative real numbers, but we demand to bank check which of these nonnegative real numbers are outputs of the office. To exercise this, nosotros will evidence that any nonnegative number is an output of this role. If , and then

Hence, the range of this function on is the set of nonnegative existent numbers, given past .

We can apply different algebraic techniques and the properties of the function to determine its domain and range. However, information technology is often easier to do this by using a sketch. Consider the post-obit sketch of .

In the sketch of whatsoever function, a point on the curve has the form , where is the input of the function and is the output. In other words, the -coordinate of every signal on the curve tells usa an input of the function and the -coordinate tells us an output of the function.

Therefore, nosotros can use the graph of a function to determine its domain and range. To determine the domain of this part, we want to find the -coordinate of every signal on the bend. Nosotros tin practice this by because which vertical lines intersect the bend.

For instance, if we sketch the vertical line , we encounter this intersects our curve at the point . Hence, 2 is in the domain of our part and 3 is in its range. To decide the total domain of our function, we need to do this with every possible vertical line. We tin can see any vertical line, , will intersect this curve. In item, for , we have the following:

Since the graph of has a solid dot at , we know the function is defined at this point. Then, the vertical line intersects the bend and . Therefore, since all vertical lines intersect the bend, its domain is .

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

For instance, the line intersects the curve at the point , and so 1 is in the range of this function. We tin can also meet that the line 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 , the line volition not intersect our bend and all other horizontal lines do intersect our curve. Hence, the range of this function is .

Before we talk over finding the domain and range of a piecewise-defined function, allow'due south kickoff 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 part'due south domain, chosen a subdomain.

The equation of a piecewise role is written with a curly subclass to indicate that it is comprised of more than one subfunction. An example of a piecewise role is where when and when .

In a piecewise-defined function, the possible inputs of the function are given by the subdomains, in this example and . Therefore, to find all the possible inputs of this function, we will demand to take the union of all the subdomains. For this piecewise function, we tin can take inputs of and also ; combining these, we can run into this is whatsoever existent value of , and so its domain is .

To find the range of a piecewise office, nosotros tin can instead consider the range of each subfunction over its subdomain. Therefore, to observe the range of , we consider the range of each subfunction separately.

First, when . Therefore, if nosotros input a value of into the function, we get

Hence, all values of are in the range of this subfunction.

Second, when . Adding 1 to both sides of the inequality of our subdomain gives us . Hence, when , . This is not enough to determine the range of this subfunction; we demand to determine which values our subfunction achieves. To exercise this, nosotros volition let so that ; this means

Therefore, all values of are in the range of this subfunction. Combining the ranges of each subfunction, we get that the range of is and ; nosotros tin can correspond this in interval notation as .

We tin summarize the results shown in the example above as follows.

Definition: Domain and Range of a Piecewise Function

The domain of a piecewise-defined function is the union of its subdomains.

The range of a piecewise-divers function is the matrimony of the ranges of each subfunction over its subdomain.

Let'south see some examples of how to detect the domain and range of a piecewise-defined role from its graph.

Example 1: Determining the Domain and Range of a Piecewise-Defined Function given Its Graph

Decide the domain and the range of the function

Answer

Nosotros recall that the domain of a office is the set of all input values of the role and the range of a function is the set of all possible outputs of the office, given its domain.

We can decide both of these from the graph of the function. Call back, any bespeak on the curve is in the form , where will be in the domain of and will be in the range of .

To notice the domain of , we need to determine the -coordinates of all points on the curve.

Consider the following vertical lines.

In diagram (one), we can see that any vertical line for intersects the bend. Similarly, in diagram (2), we can run into that any vertical line for intersects the line. So, all of these values of must be in the domain of this function. Nosotros need to cheque if is in the domain of this part; we tin cheque this by sketching the line .

Since our bend has hollow circles on the line , it is not divers at this value of ; therefore, 0 is not in the domain of . Hence, the domain of this function is all real values of non equal to 0, which we can write in fix notation as .

Information technology is worth noting that we can verify that 0 is not in the domain of past considering the subdomains of the function, and , which both practice not include 0. The union of these subdomains is also the domain of the function, .

To find the range of this role, we could consider which horizontal lines intersect the graph. However, in this example, we can find the range by because the coordinates of the points on the graph.

We tin can meet that if , and so . Similarly, if , then . This means the only possible outputs of our office are 6 and , and then the range of this function is .

Hence, the domain is , and the range is .

Example 2: Determining the Range of a Piecewise-Defined Function given Its Graph

Find the range of the function

Answer

Nosotros retrieve that the range of a function is the fix of all possible outputs of the function, given its domain. To notice the range of this function, we can consider which horizontal lines intersect the graph.

In diagram (1), we tin can run into that the highest output of the function is . In diagram (2), nosotros can encounter that the lowest output of the function is . All the values between these are possible outputs giving us the range .

Hence, the range is .

Case 3: Determining the Range of Piecewise-Defined Functions from Their Graphs

Determine the range of the function represented by the given graph.

Respond

We recollect that the range of a function is the ready of all possible outputs of the office, given its domain. Remember, whatsoever point on our graph will exist in the class , where will be in the range of the office. Therefore, we can observe the range of this office by determining the -coordinates of the points on its graph.

In diagram (1), since our graph has a solid dot at , we can see that the lowest output of the part is . In the second diagram, nosotros can meet that the largest output of the function is 7.

We tin can see that any horizontal line between these values also intersects the curve, so the range of this part is any value between and 7, inclusive. In set note, this is .

Hence, the range is .

In our next example, nosotros will see how to determine the domain of a piecewise-defined function without being given its graph.

Example 4: Determining the Domain of a Piecewise-Defined Function

Determine the domain of the part

Respond

We recall that the domain of a function is the set of all input values of the role, and for a piecewise-divers function, information technology is the marriage of its subdomains.

For this function, the subdomains are and . Nosotros want to take the union of these two sets to find the domain of :

Hence, the domain is .

In our final example, we will see how to decide both the domain and the range of a piecewise-defined function without beingness given its graph.

Example v: Determining the Domain and the Range of a Piecewise Function

Decide the domain and range of the function

Answer

Nosotros recall that the domain of a function is the set of all input values of the function, and for a piecewise-divers role, it is the union of its subdomains.

To find the spousal relationship of the subdomains, nosotros volition start by writing them in terms of sets. First, is the same every bit . Second, is the same as .

Therefore, the domain is the union of these sets:

The range of a office is the ready of all possible outputs of the function, given its domain. For a piecewise-defined function, this will be the range of the subfunctions over their subdomains. And then, nosotros can decide the range of this office by considering each subfunction separately.

First, if , since , we can cancel the shared factor of :

We can then sketch this subfunction.

It is the line with the point when removed. The range of this subfunction is all of the possible outputs. The only horizontal line that does not intersect this line is , so the range of this subfunction is .

The 2nd subfunction is the constant role on the domain . Since the output is constant, its range is .

Taking the marriage of the ranges of the subfunctions gives us

It is worth noting that nosotros could likewise sketch the second subfunction on the same graph to fully sketch . The second subfunction is just defined when , so it consists of a single point. Nosotros have , and so we add together the point to our sketch.

We can and then see that is function .

Hence, the domain is and the range is .

Permit'south end by recapping some of the of import points of this explainer.

Fundamental Points

The domain of a piecewise-defined function is the union of its subdomains.
The range of a piecewise-defined function is the union of the ranges of each subfunction over its subdomain.
We can find the domain of a function from its graph by because the intersections of the curve with vertical lines.
We tin notice the range of a part from its graph past considering the intersections of the curve with horizontal lines.