Lesson Explainer: The Domain and the Range of a Radical Function | Nagwa Lesson Explainer: The Domain and the Range of a Radical Function | Nagwa

Lesson Explainer: The Domain and the Range of a Radical Function Mathematics • Second Year of Secondary School

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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 𝑓(𝑥) is a set of all possible values 𝑥 such that the expression 𝑓(𝑥) is defined.

The range of a function 𝑓(𝑥) is a set of all possible values the expression 𝑓(𝑥) can take, when 𝑥 is any number from the domain of the function.

For example, consider the function 𝑓(𝑥)=𝑥. If there is a real number 𝑦 satisfying 𝑦=𝑥, it must be the case that 𝑥=𝑦. We know that the square of any real number is not negative; thus 𝑥 must be nonnegative. This tells us that the domain of the square root function is 𝑥0, which is expressed in the interval notation as [0,[.

The graph of the square root function is shown below.

In the figure above, the square root function is graphed over the interval [0,2]. Although the graph of the function appears to become flatter for larger values of 𝑥, it continues to increase without a bound. We can observe this effect by graphing the square root function over a larger interval, [0,1000].

We can see that the 𝑦-value of the graph keeps increasing for larger values of 𝑥. In fact, given any large positive number 𝑦, we know that 𝑦=𝑦. Also, we know that 0=0. Hence, the range of the square root function is [0,[.

Theorem: Domain and Range of the Square Root Function

The domain and range of the square root function, 𝑓(𝑥)=𝑥, is [0,[.

More generally, the domain of a composite square root function 𝑔(𝑥) can be identified by finding the values of 𝑥 satisfying 𝑔(𝑥)0.

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 𝑓(𝑥)=𝑥.

  1. Find the domain of 𝑓(𝑥).
  2. Find the range of 𝑓(𝑥).

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, 𝑥0.

This leads to 𝑥0, which is ],0] in interval notation.

The domain of 𝑓(𝑥) is ],0].

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 𝑥 is [0,[. In other words, for any number 𝑦 in the interval [0,[, we can find some number 𝑥 that satisfies 𝑦=𝑥. This means that the number 𝑥 satisfies 𝑓(𝑥)=(𝑥)=𝑥=𝑦.

Hence, any number in the interval [0,[ is a possible function value of 𝑓(𝑥)=𝑥.

The range of 𝑓(𝑥) is [0,[.

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 𝑓(𝑥)=7𝑥7.

Answer

Recall that the domain of a function 𝑔(𝑥) is the set of 𝑥-values satisfying 𝑔(𝑥)0.

In this example, 𝑔(𝑥)=7𝑥7. Hence, the domain of this function is the set of 𝑥-values such that 7𝑥70.

Rearranging this inequality leads to 𝑥1, which is written as [1,[ in interval notation.

The domain of 𝑓(𝑥)=7𝑥7 is [1,[.

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 𝑓(𝑥)=12𝑥?

Answer

Let us use the domain and the range of 𝑓(𝑥)=12𝑥 to identify the graph. We know that the domain of 𝑔(𝑥) is the set of 𝑥-values satisfying 𝑔(𝑥)0.

In this example, 𝑔(𝑥)=12𝑥. Hence, the domain of this function is the set of 𝑥-values such that 12𝑥0.

Rearranging this inequality leads to 𝑥12, which is written as ,12 in interval notation. Hence, the domain of 𝑓(𝑥) is ,12.

Recall that the range of the square root function 𝑥 is [0,[. We have observed that 𝑓(𝑥)=𝑔(𝑥) where 𝑔(𝑥)=12𝑥. Since the range of 𝑔(𝑥)=12𝑥 is all real numbers, 𝑔(𝑥) must have the same range as 𝑥. This leads to the range of 𝑓(𝑥), which is [0,[.

Let us identify which one of the graphs represents a function whose domain is ,12 and whose range is [0,[. 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.

  1. Domain: 12,, range: [0,[
  2. Domain: ,12, range: ],0[
  3. Domain: ],[, range: ],[
  4. Domain: ,12, range: [0,[
  5. Domain: 12,, range: [0,[

Therefore, the only possible match for 𝑓(𝑥)=12𝑥 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 𝑓(𝑥)=𝑥+7𝑥5.

Answer

In the given function, we observe two types of restrictions to the domain.

  • Square root: there are two expressions 𝑥+7 and 𝑥5. We recall that the square root function cannot take a negative number.
  • Denominator: this is the expression 𝑥5. We recall that the denominator of a fraction cannot equal zero.

We begin by considering the restrictions imposed by the square roots. For 𝑥+7 to be well defined, we need 𝑥+70. This leads to 𝑥7, which is the interval [7,[.

For 𝑥5 to be well defined, we need 𝑥50, which leads to 𝑥5, or [5,[.

Finally, let us consider the denominator. Since the denominator cannot equal zero, we need to exclude the case when 𝑥5=0. Squaring both sides of this equation gives 𝑥5=0, leading to 𝑥=5. Hence, we need to impose 𝑥5.

The domain of this function is the set of 𝑥-values satisfying all three conditions: [7,[,[5,[,𝑥5.

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 [7,[, the green highlight represents the interval [5,[, and the red X represents the restriction 𝑥5. We can draw an interval that satisfies the intersection of these restrictions simultaneously.

Hence, the domain of 𝑓(𝑥) is ]5,[.

Example 5: Finding the Domain of Rational Functions

Find the domain of 𝑓(𝑥)=𝑥19𝑥𝑥3.

Answer

In the given function, we observe two types of restrictions to the domain.

  • Square root: there are the three expressions 𝑥1, 9𝑥, and 𝑥3. We recall that the square root function cannot take a negative number.
  • Denominator: this is the expression 9𝑥𝑥3. We recall that the denominator of a fraction cannot equal zero.

We begin by considering the restrictions imposed by the square roots. For 𝑥1 to be well defined, we need 𝑥10. This leads to 𝑥1, which is the interval [1,[.

Similarly, the expressions 9𝑥 and 𝑥3 lead to the intervals ],9] and [3,[ respectively.

Next, let us consider the denominator. Since the denominator cannot be equal to zero, we need to exclude the case when 9𝑥𝑥3=0.

We can add 𝑥3 to both sides of this equation to obtain 9𝑥=𝑥3.

Squaring both sides leads to 9𝑥=𝑥312=2𝑥𝑥=6.

Hence, we need to impose 𝑥6.

We have found four domain restrictions for 𝑓(𝑥): [1,[,],9],[3,[,𝑥6.

Let us draw a number line to visualize how these restrictions interact.

In the diagram above, the purple highlight represents the interval [1,[, the green highlight represents the interval ],9], the blue highlight represents [3,[, and the red X represents the restriction 𝑥6. We can draw the set satisfying all four restrictions simultaneously.

Hence, the domain of 𝑓(𝑥) is [3,9]{6}.

In previous examples, we consider the domain and range of square root functions with linear expressions, 𝑎𝑥+𝑏. 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 𝑓(𝑥)=4|𝑥5|.

  1. Find the domain of 𝑓(𝑥).
  2. Find the range of 𝑓(𝑥).

Answer

Part 1

Let us find the domain of the given function. We know that the domain of 𝑔(𝑥) is the set of 𝑥-values satisfying 𝑔(𝑥)0.

In the function 𝑓(𝑥), the expression 4|𝑥5| is under the square root. Hence, we need to find the values of 𝑥 such that 4|𝑥5|0.

To solve an inequality with an absolute value, we begin by isolating the absolute value on one side of the inequality. Adding |𝑥5| to both sides of the inequality, we obtain 4|𝑥5|.

We can consider this in two separate situations. First, consider the case that 𝑥5 is nonnegative. Since the absolute value does not change a nonnegative number, the inequality is the same as 4𝑥5.

Second, consider the case when 𝑥5 is negative. Since the absolute value takes away a negative sign, 𝑥5 needs to be at least 4 to satisfy the inequality 4|𝑥5|. In other words, 𝑥54. Together, 4|𝑥5| is the same as 4𝑥54.

Adding 5 to the inequality leads to 1𝑥9.

The domain of 𝑓(𝑥) is [1,9].

Part 2

Let us find the range of 𝑓(𝑥). 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 𝑓(𝑥) 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 4=|𝑥5|.

This is possible when 𝑥=1 or 𝑥=9. This means that the smallest possible value of 𝑓(𝑥) is 0.

To find the largest possible value of 𝑓(𝑥), we note that |𝑥5| is a nonnegative number. Since the expression inside the square root is 4|𝑥5|, the function would have the largest value when |𝑥5|=0, which happens when 𝑥=5. In this case, the function value is 4=2. So, the largest possible value of 𝑓(𝑥) is 2.

To finally conclude that the range of this function is [0,2], we need to know that all values between 0 and 2 are possible. If 𝑦 is any value between 0 and 2, let us find a number 𝑥 such that 𝑓(𝑥)=𝑦. We have 𝑦=4|𝑥5|.

Squaring both sides of the equation, 𝑦=4|𝑥5|.

Rearranging the equation, |𝑥5|=4𝑦.

As we have done previously, we can split this equation into the two cases, depending on the sign of 𝑥5. But, since we just need to find one possible value 𝑥, let us just take 𝑥5>0. In this case, we have 𝑥5=4𝑦, which leads to 𝑥=9𝑦. Let us check if 𝑓9𝑦=𝑦. Substituting 𝑥=9𝑦 into the function 𝑓(𝑥), we obtain 𝑓9𝑦=4|(9𝑦)5|=4|4𝑦|.

Since 0𝑦2, we know that 4𝑦>0, so we can ignore the absolute value in ||4𝑦||. Continuing from above: =4(4𝑦)=44+𝑦=𝑦.

Since 𝑦>0, 𝑦=𝑦. Hence, we have shown that, for any 0𝑦2, we have 𝑥=9𝑦 that satisfies 𝑓(𝑥)=𝑦.

The range of 𝑓(𝑥) is [0,2].

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, 𝑥.

Unlike the square root function, we note that the function extends to the left and the right side of the 𝑦-axis, indicating that the cube root can take any real numbers. We also note that the 𝑦-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, 𝑓(𝑥)=𝑥, 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, 𝑎𝑥+𝑏.

Example 7: Finding the Domain of a Cubic Root Function

Determine the domain of the function 𝑓(𝑥)=4𝑥+3.

Answer

Recall that the domain and the range of the cube root function 𝑥 is ],[. In other words, the cube root function does not impose any domain restriction. Since the expression, 4𝑥+3, under the cube root does not have any domain restriction, there are no restrictions to possible 𝑥-values for this function.

Hence, the domain of 𝑓(𝑥)=4𝑥+3 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 𝑓(𝑥)=1252𝑥+3.

  1. Find the domain of 𝑓(𝑥).
  2. Find the range of 𝑓(𝑥).

Answer

Part 1

Let us find the domain of 𝑓(𝑥). We recall the domain restrictions for the square and the cube roots.

  • The square root function, 𝑥, has the domain [0,[.
  • The cube root function, 𝑥, 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 2𝑥+3. Setting the expression 2𝑥+3 to be nonnegative, 2𝑥+30.

Rearranging this inequality leads to 𝑥1.5.

The domain of 𝑓(𝑥) is [1.5,[.

Part 2

Let us find the range of 𝑓(𝑥). First, consider the expression 2𝑥+3. Recall that the range of the square root function 𝑥 is [0,[. Note that 2𝑥+3 can be written as 2𝑥+3=𝑔(2𝑥+3),𝑔(𝑥)=𝑥.where

Since the range of 2𝑥+3 is all real numbers, the range of 2𝑥+3 must be the same as the range of 𝑔(𝑥). Thus, the range of 2𝑥+3 is [0,[. This tells us that the expression 2𝑥+3 will output nonnegative values. Then, possible values for 𝑓(𝑥)=1252𝑥+3 can be written as 125𝑎,𝑎0.forsome

The largest possible value of this happens when 𝑎=0, which gives us 125=5. Since the largest function value is 5 and we know that the cube root function, 𝑥, tends to to the left of 𝑦-axis, we would like to conclude that the range of the function is ],5]. Let us carefully justify this conclusion.

If 𝑦 is some number satisfying 𝑦5, we need to show that 𝑦=125𝑎𝑎0.forsome

We can raise both sides of the equation to the cubic power to obtain 𝑦=125𝑎.

Rearranging this equation leads to 𝑎=125𝑦.

Since 𝑦5, we have 𝑦125, which means 𝑎0. This tells us that 𝑦 is a possible value of the function as long as 𝑦5.

The range of 𝑓(𝑥) is ],5].

Let us finish by recapping a few important concepts from this explainer.

Key Points

  • The domain and range of the square root function 𝑓(𝑥)=𝑥 is [0,[.
    More generally, the domain of a composite square root function 𝑔(𝑥) can be identified by finding the values of 𝑥 satisfying 𝑔(𝑥)0.
  • The domain and range of the cube root function, 𝑓(𝑥)=𝑥, are all real numbers. This is denoted as ],[ or .
  • If the range of the function 𝑔(𝑥) is all real numbers, then the range of 𝑔(𝑥) or 𝑔(𝑥) is [0,[ or respectively.

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