Loading [MathJax]/jax/output/HTML-CSS/jax.js
Skip to main content
Library homepage
 

Text Color

Text Size

 

Margin Size

 

Font Type

Enable Dyslexic Font
Mathematics LibreTexts

17.8: Use Radicals in Functions

( \newcommand{\kernel}{\mathrm{null}\,}\)

Learning Objectives

By the end of this section, you will be able to:

  • Evaluate a radical function
  • Find the domain of a radical function
  • Graph radical functions

Before you get started, take this readiness quiz.

  1. Solve: 12x0.
    If you missed this problem, review Example 2.50.
  2. For f(x)=3x4, evaluate f(2),f(1),f(0).
    If you missed this problem, review Example 3.48.
  3. Graph f(x)=x. State the domain and range of the function in interval notation.
    If you missed this problem, review Example 3.56.

Evaluate a Radical Function

In this section we will extend our previous work with functions to include radicals. If a function is defined by a radical expression, we call it a radical function.

  • The square root function is f(x)=x.
  • The cube root function is f(x)=3x.
Definition 17.8.1: radical function

A radical function is a function that is defined by a radical expression.

To evaluate a radical function, we find the value of f(x) for a given value of x just as we did in our previous work with functions.

Example 17.8.1

For the function f(x)=2x1, find

  1. f(5)
  2. f(2)

Solution:

a.

f(x)=2x1

To evaluate f(5), substitute 5 for x.

f(5)=251

Simplify.

f(5)=9

Take the square root.

f(5)=3

b.

f(x)=2x1

To evaluate f(2), substitute 2 for x.

f(2)=2(2)1

Simplify.

f(2)=5

Since the square root of a negative number is not a real number, the function does not have a value at x=2.

Exercise 17.8.1

For the function f(x)=3x2, find

  1. f(6)
  2. f(0)
Answer
  1. f(6)=4
  2. no value at x=0
Exercise 17.8.2

For the function g(x)=5x+5, find

  1. g(4)
  2. g(3)
Answer
  1. g(4)=5
  2. no value at f(3)

We follow the same procedure to evaluate cube roots.

Example 17.8.2

For the function g(x)=3x6, find

  1. g(14)
  2. g(2)

Solution:

a.

g(x)=3x6

To evaluate g(14), substitute 14 for x.

g(14)=3146

Simplify.

g(14)=38

Take the cube root.

g(14)=2

b.

g(x)=3x6

To evaluate g(2), substitute 2 for x.

g(2)=326

Simplify.

g(2)=38

Take the cube root.

g(2)=2

Exercise 17.8.3

For the function g(x)=33x4, find

  1. g(4)
  2. g(1)
Answer
  1. g(4)=2
  2. g(1)=1
Exercise 17.8.4

For the function h(x)=35x2, find

  1. h(2)
  2. h(5)
Answer
  1. h(2)=2
  2. h(5)=3

The next example has fourth roots.

Example 17.8.3

For the function f(x)=45x4, find

  1. f(4)
  2. f(12)

Solution:

a.

f(x)=45x4

To evaluate f(4), substitute 4 for x.

f(4)=4544

Simplify.

f(4)=416

Take the fourth root.

f(4)=2

b.

f(x)=45x4

To evaluate f(12), substitute 12 for x.

f(12)=45(12)4

Simplify.

f(12)=464

Since the fourth root of a negative number is not a real number, the function does not have a value at x=12.

Exercise 17.8.5

For the function f(x)=43x+4, find

  1. f(4)
  2. f(1)
Answer
  1. f(4)=2
  2. f(1)=1
Exercise 17.8.6

For the function g(x)=45x+1, find

  1. g(16)
  2. g(3)
Answer
  1. g(16)=3
  2. g(3)=2

Find the Domain of a Radical Function

To find the domain and range of radical functions, we use our properties of radicals. For a radical with an even index, we said the radicand had to be greater than or equal to zero as even roots of negative numbers are not real numbers. For an odd index, the radicand can be any real number. We restate the properties here for reference.

Properties of na

When n is an even number and:

  • a0, then na is a real number.
  • a<0, then na is not a real number.

When n is an odd number, na is a real number for all values of a.

So, to find the domain of a radical function with even index, we set the radicand to be greater than or equal to zero. For an odd index radical, the radicand can be any real number.

Domain of a Radical Function

When the index of the radical is even, the radicand must be greater than or equal to zero.

When the index of the radical is odd, the radicand can be any real number.

Example 17.8.4

Find the domain of the function, f(x)=3x4. Write the domain in interval notation.

Solution:

Since the function, f(x)=3x4 has a radical with an index of 2, which is even, we know the radicand must be greater than or equal to 0. We set the radicand to be greater than or equal to 0 and then solve to find the domain.

Solve.

3x403x4x43

The domain of f(x)=3x4 is all values x43 and we write it in interval notation as [43,).

Exercise 17.8.7

Find the domain of the function, f(x)=6x5. Write the domain in interval notation.

Answer

[56,)

Exercise 17.8.8

Find the domain of the function, f(x)=45x. Write the domain in interval notation.

Answer

(,45]

Example 17.8.5

Find the domain of the function, g(x)=6x1. Write the domain in interval notation.

Solution:

Solve the function, g(x)=6x1 has a radical with an index of 2, which is even, we know the radicand must be greater than or equal to 0.

The radicand cannot be zero since the numerator is not zero.

For 6x1 to be greater than zero, the denominator must be positive since the numerator is positive. We know a positive divided by a positive is positive.

We set x1>0 and solve.

x1>0

Solve.

x>1

Also, since the radicand is a fraction, we must realize that the denominator cannot be zero.

We solve x1=0 to find the value that must be eliminated from the domain.

x1=0

Solve.

x=1 so x/neq1 in the domain.

Putting this together we get the domain is x>1 and we write it as (1,).

Exercise 17.8.9

Find the domain of the function, f(x)=4x+3. Write the domain in interval notation.

Answer

(3,)

Exercise 17.8.10

Find the domain of the function, h(x)=9x5. Write the domain in interval notation.

Answer

(5,)

The next example involves a cube root and so will require different thinking.

Example 17.8.6

Find the domain of the function, f(x)=32x2+3. Write the domain in interval notation.

Solution:

Since the function, f(x)=32x2+3 has a radical with an index of 3, which is odd, we know the radicand can be any real number. This tells us the domain is any real number. In interval notation, we write (,).

The domain of f(x)=32x2+3 is all real numbers and we write it in interval notation as (,).

Exercise 17.8.11

Find the domain of the function, f(x)=33x21. Write the domain in interval notation.

Answer

(,)

Exercise 17.8.12

Find the domain of the function, g(x)=35x4. Write the domain in interval notation.

Answer

(,)

Graph Radical Functions

Before we graph any radical function, we first find the domain of the function. For the function, f(x)=x, the index is even, and so the radicand must be greater than or equal to 0.

This tells us the domain is x0 and we write this in interval notation as [0,).

Previously we used point plotting to graph the function, f(x)=x. We chose x-values, substituted them in and then created a chart. Notice we chose points that are perfect squares in order to make taking the square root easier.

The figure shows the square root function graph on the x y-coordinate plane. The x-axis of the plane runs from 0 to 7. The y-axis runs from 0 to 7. The function has a starting point at (0, 0) and goes through the points (1, 1) and (4, 2). A table is shown beside the graph with 3 columns and 5 rows. The first row is a header row with the expressions “x”, “f (x) = square root of x”, and “(x, f (x))”. The second row has the numbers 0, 0, and (0, 0). The third row has the numbers 1, 1, and (1, 1). The fourth row has the numbers 4, 2, and (4, 2). The fifth row has the numbers 9, 3, and (9, 3).
Figure 8.7.1

Once we see the graph, we can find the range of the function. The y-values of the function are greater than or equal to zero. The range then is [0,).

Example 17.8.7

For the function f(x)=x+3,

  1. find the domain
  2. graph the function
  3. use the graph to determine the range

Solution:

  1. Since the radical has index 2, we know the radicand must be greater than or equal to zero. If x+30, then x3. This tells us the domain is all values x3 and written in interval notation as [3,).
  2. To graph the function, we choose points in the interval [3,) that will also give us a radicand which will be easy to take the square root.
The figure shows a square root function graph on the x y-coordinate plane. The x-axis of the plane runs from negative 3 to 3. The y-axis runs from 0 to 7. The function has a starting point at (negative 3, 0) and goes through the points (negative 2, 1) and (1, 2). A table is shown beside the graph with 3 columns and 5 rows. The first row is a header row with the expressions “x”, “f (x) = square root of the quantity x plus 3”, and “(x, f (x))”. The second row has the numbers negative 3, 0, and (negative 3, 0). The third row has the numbers negative 2, 1, and (negative 2, 1). The fourth row has the numbers 1, 2, and (1, 2). The fifth row has the numbers 6, 3, and (6, 3).
Figure 8.7.2

c. Looking at the graph, we see the y-values of the function are greater than or equal to zero. The range then is [0,).

Exercise 17.8.13

For the function f(x)=x+2,

  1. find the domain
  2. graph the function
  3. use the graph to determine the range
Answer
  1. domain: [2,)

  2. The figure shows a square root function graph on the x y-coordinate plane. The x-axis of the plane runs from negative 2 to 6. The y-axis runs from 0 to 8. The function has a starting point at (negative 2, 0) and goes through the points (negative 1, 1) and (2, 2).
    Figure 8.7.3
  3. range: [0,)
Exercise 17.8.14

For the function f(x)=x2,

  1. find the domain
  2. graph the function
  3. use the graph to determine the range
Answer
  1. domain: [2,)

  2. The figure shows a square root function graph on the x y-coordinate plane. The x-axis of the plane runs from 0 to 8. The y-axis runs from 0 to 6. The function has a starting point at (2, 0) and goes through the points (3, 1) and (6, 2).
    Figure 8.7.4
  3. range: [0,)

In our previous work graphing functions, we graphed f(x)=x3 but we did not graph the function f(x)=3x. We will do this now in the next example.

Example 17.8.8

For the function, f(x)=3x,

  1. find the domain
  2. graph the function
  3. use the graph to determine the range

Solution:

a. Since the radical has index 3, we know the radicand can be any real number. This tells us the domain is all real numbers and written in interval notation as (,)

b. To graph the function, we choose points in the interval (,) that will also give us a radicand which will be easy to take the cube root.

The figure shows the cube root function graph on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis runs from negative 10 to 10. The function has a center point at (0, 0) and goes through the points (1, 1), (negative 1, negative 1), (8, 2), and (negative 8, negative 2). A table is shown beside the graph with 3 columns and 6 rows. The first row is a header row with the expressions “x”, “f (x) = cube root of x”, and “(x, f (x))”. The second row has the numbers negative 8, negative 2, and (negative 8, negative 2). The third row has the numbers negative 1, negative 1, and (negative 1, negative 1). The fourth row has the numbers 0, 0, and (0, 0). The fifth row has the numbers 1, 1, and (1, 1). The sixth row has the numbers 8, 2, and (8, 2).
Figure 8.7.5

c. Looking at the graph, we see the y-values of the function are all real numbers. The range then is (,).

Exercise 17.8.15

For the function f(x)=3x,

  1. find the domain
  2. graph the function
  3. use the graph to determine the range
Answer
  1. domain: (,)

  2. The figure shows a cube root function graph on the x y-coordinate plane. The x-axis of the plane runs from negative 2 to 2. The y-axis runs from negative 2 to 2. The function has a center point at (0, 0) and goes through the points (1, negative 1) and (negative 1, 1).
    Figure 8.7.6
  3. range: (,)
Exercise 17.8.16

For the function f(x)=3x2,

  1. find the domain
  2. graph the function
  3. use the graph to determine the range
Answer
  1. domain: (,)

  2. The figure shows a cube root function graph on the x y-coordinate plane. The x-axis of the plane runs from negative 1 to 5. The y-axis runs from negative 3 to 3. The function has a center point at (2, 0) and goes through the points (1, negative 1) and (3, 2).
    Figure 8.7.7
  3. range: (,)

Access these online resources for additional instruction and practice with radical functions.

  • Domain of a Radical Function
  • Domain of a Radical Function 2
  • Finding Domain of a Radical Function

Key Concepts

  • Properties of na
    • When n is an even number and:
      a0, then na is a real number.
      a<0, then na is not a real number.
    • When n is an odd number, na is a real number for all values of a.
  • Domain of a Radical Function
    • When the index of the radical is even, the radicand must be greater than or equal to zero.
    • When the index of the radical is odd, the radicand can be any real number.

Glossary

radical function
A radical function is a function that is defined by a radical expression.

17.8: Use Radicals in Functions is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

  • Was this article helpful?

Support Center

How can we help?