In this explainer, we will learn how to find the domain and the range of a radical function either from its graph or from its defining rule.
In particular, we will focus on the domain and range of functions involving the square and the cube roots.
Let us begin by recalling the definitions of domain and range of a function.
Theorem: Domain and Range of a Function
The domain of a function is a set of all possible values such that the expression is defined.
The range of a function is a set of all possible values the expression can take, when is any number from the domain of the function.
For example, consider the function . If there is a real number satisfying , it must be the case that . We know that the square of any real number is not negative; thus must be nonnegative. This tells us that the domain of the square root function is , which is expressed in the interval notation as .
The graph of the square root function is shown below.
In the figure above, the square root function is graphed over the interval . Although the graph of the function appears to become flatter for larger values of , it continues to increase without a bound. We can observe this effect by graphing the square root function over a larger interval, .
We can see that the -value of the graph keeps increasing for larger values of . In fact, given any large positive number , we know that . Also, we know that . Hence, the range of the square root function is .
Theorem: Domain and Range of the Square Root Function
The domain and range of the square root function, , is .
More generally, the domain of a composite square root function can be identified by finding the values of satisfying .
Let us consider a few examples where we identify the domain and the range of functions involving the square root.
Example 1: Finding the Domain of Root Functions
Consider the function .
- Find the domain of .
- Find the range of .
Answer
Part 1
We recall that the square root cannot take a negative number as an argument. Hence, the domain of the given function is found by setting the expression inside the square root to be greater than or equal to zero. In other words,
This leads to , which is in interval notation.
The domain of is .
Part 2
The range of a function is the set of all possible function values. We know that the range of the square root function is . In other words, for any number in the interval , we can find some number that satisfies . This means that the number satisfies
Hence, any number in the interval is a possible function value of .
The range of is .
Let us consider another example for obtaining the domain of a composite square root function.
Example 2: Finding the Domain of Root Functions
Find the domain of the function .
Answer
Recall that the domain of a function is the set of -values satisfying
In this example, . Hence, the domain of this function is the set of -values such that
Rearranging this inequality leads to , which is written as in interval notation.
The domain of is .
In the next example, we will determine the correct graph of a composite square root function by considering its domain and range.
Example 3: Identifying the Graph of a Radical Function
Which of the following is the graph of ?
Answer
Let us use the domain and the range of to identify the graph. We know that the domain of is the set of -values satisfying
In this example, . Hence, the domain of this function is the set of -values such that
Rearranging this inequality leads to , which is written as in interval notation. Hence, the domain of is .
Recall that the range of the square root function is . We have observed that where . Since the range of is all real numbers, must have the same range as . This leads to the range of , which is .
Let us identify which one of the graphs represents a function whose domain is and whose range is . From each given graph, the domain of the function corresponding to the graph is the part of the horizontal axis where the graph exists. Also, the range of the function is the part of the vertical axis where the graph exists. We obtain the domain and range of each function using its graph.
- Domain: , range:
- Domain: , range:
- Domain: , range:
- Domain: , range:
- Domain: , range:
Therefore, the only possible match for is D.
We might recall that when working with rational functions, we must be careful to restrict the domain to ensure that the function on the denominator of the expression cannot be equal to zero. Let us demonstrate how to find the domain of a function which is a ratio of two composite square root functions.
Example 4: Finding the Domain of Rational Functions
Determine the domain of the function .
Answer
In the given function, we observe two types of restrictions to the domain.
- Square root: there are two expressions and . We recall that the square root function cannot take a negative number.
- Denominator: this is the expression . We recall that the denominator of a fraction cannot equal zero.
We begin by considering the restrictions imposed by the square roots. For to be well defined, we need . This leads to , which is the interval .
For to be well defined, we need , which leads to , or .
Finally, let us consider the denominator. Since the denominator cannot equal zero, we need to exclude the case when . Squaring both sides of this equation gives , leading to . Hence, we need to impose .
The domain of this function is the set of -values satisfying all three conditions:
To see how these three restrictions interact, let us draw a number line with these restrictions.
In the diagram above, the purple highlight represents the interval , the green highlight represents the interval , and the red X represents the restriction . We can draw an interval that satisfies the intersection of these restrictions simultaneously.
Hence, the domain of is .
Example 5: Finding the Domain of Rational Functions
Find the domain of .
Answer
In the given function, we observe two types of restrictions to the domain.
- Square root: there are the three expressions , , and . We recall that the square root function cannot take a negative number.
- Denominator: this is the expression . We recall that the denominator of a fraction cannot equal zero.
We begin by considering the restrictions imposed by the square roots. For to be well defined, we need . This leads to , which is the interval .
Similarly, the expressions and lead to the intervals and respectively.
Next, let us consider the denominator. Since the denominator cannot be equal to zero, we need to exclude the case when
We can add to both sides of this equation to obtain
Squaring both sides leads to
Hence, we need to impose .
We have found four domain restrictions for :
Let us draw a number line to visualize how these restrictions interact.
In the diagram above, the purple highlight represents the interval , the green highlight represents the interval , the blue highlight represents , and the red X represents the restriction . We can draw the set satisfying all four restrictions simultaneously.
Hence, the domain of is .
In previous examples, we consider the domain and range of square root functions with linear expressions, . In the next example, we will find the domain and the range of a composite square root function where the expression inside the square root involves the absolute value function.
Example 6: Finding the Domain of Rational Functions
Consider the function .
- Find the domain of .
- Find the range of .
Answer
Part 1
Let us find the domain of the given function. We know that the domain of is the set of -values satisfying
In the function , the expression is under the square root. Hence, we need to find the values of such that
To solve an inequality with an absolute value, we begin by isolating the absolute value on one side of the inequality. Adding to both sides of the inequality, we obtain
We can consider this in two separate situations. First, consider the case that is nonnegative. Since the absolute value does not change a nonnegative number, the inequality is the same as
Second, consider the case when is negative. Since the absolute value takes away a negative sign, needs to be at least to satisfy the inequality . In other words, . Together, is the same as
Adding 5 to the inequality leads to
The domain of is .
Part 2
Let us find the range of . The range of a function is the set of all possible values of the function. We can obtain the range of a function by considering what are the largest and the smallest possible values of the function. Since any function value is a square root of a number, we know that it cannot be negative. Thus, 0 would be the smallest possible function value if it is possible. For this, we need the expression under the square root to be equal to zero. This leads to
This is possible when or . This means that the smallest possible value of is 0.
To find the largest possible value of , we note that is a nonnegative number. Since the expression inside the square root is , the function would have the largest value when , which happens when . In this case, the function value is . So, the largest possible value of is 2.
To finally conclude that the range of this function is , we need to know that all values between 0 and 2 are possible. If is any value between 0 and 2, let us find a number such that . We have
Squaring both sides of the equation,
Rearranging the equation,
As we have done previously, we can split this equation into the two cases, depending on the sign of . But, since we just need to find one possible value , let us just take . In this case, we have which leads to . Let us check if . Substituting into the function , we obtain
Since , we know that , so we can ignore the absolute value in . Continuing from above:
Since , . Hence, we have shown that, for any , we have that satisfies .
The range of is .
We have considered many examples on domain and range of functions involving the square root. Unlike the square root, the cube root function does not impose any restrictions on the domain or the range. Following is the graph of the cube root function, .
Unlike the square root function, we note that the function extends to the left and the right side of the -axis, indicating that the cube root can take any real numbers. We also note that the -values tend to positive or negative infinity as we move to the left or the right. This indicates that the range of the cube root function is all real numbers.
Theorem: Domain and Range of the Cube Root Function
The domain and range of the cube root function, , are all real numbers. This is denoted as or .
In the next example, we will identify the domain of a cube root function where the expression under the cube root is linear, .
Example 7: Finding the Domain of a Cubic Root Function
Determine the domain of the function .
Answer
Recall that the domain and the range of the cube root function is . In other words, the cube root function does not impose any domain restriction. Since the expression, , under the cube root does not have any domain restriction, there are no restrictions to possible -values for this function.
Hence, the domain of is all real numbers, .
In our final example, we will find the domain and the range of a function that involves both the square and the cube roots.
Example 8: Finding the Domain of Rational Functions
Consider the function .
- Find the domain of .
- Find the range of .
Answer
Part 1
Let us find the domain of . We recall the domain restrictions for the square and the cube roots.
- The square root function, , has the domain .
- The cube root function, , has no domain restrictions. The domain of the cube root function is all real numbers, or .
Since the cube root does not impose any domain restriction, we only need to consider the restriction from the square root expression . Setting the expression to be nonnegative,
Rearranging this inequality leads to .
The domain of is .
Part 2
Let us find the range of . First, consider the expression . Recall that the range of the square root function is . Note that can be written as
Since the range of is all real numbers, the range of must be the same as the range of . Thus, the range of is . This tells us that the expression will output nonnegative values. Then, possible values for can be written as
The largest possible value of this happens when , which gives us . Since the largest function value is 5 and we know that the cube root function, , tends to to the left of -axis, we would like to conclude that the range of the function is . Let us carefully justify this conclusion.
If is some number satisfying , we need to show that
We can raise both sides of the equation to the cubic power to obtain
Rearranging this equation leads to
Since , we have , which means . This tells us that is a possible value of the function as long as .
The range of is .
Let us finish by recapping a few important concepts from this explainer.
Key Points
- The domain and range of the square root function is .
More generally, the domain of a composite square root function can be identified by finding the values of satisfying . - The domain and range of the cube root function, , are all real numbers. This is denoted as or .
- If the range of the function is all real numbers, then the range of or is or respectively.
