Use Radicals in Functions - Intermediate Algebra

8.7 Use Radicals in Functions

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

Be Prepared 8.19

Before you get started, take this readiness quiz.

Solve: $1 - 2 x \geq 0 .$
If you missed this problem, review Example 2.50.

Be Prepared 8.20

For $f (x) = 3 x - 4,$ evaluate $f (2), f (−1), f (0) .$
If you missed this problem, review Example 3.48.

Be Prepared 8.21

Graph $f (x) = \sqrt{x} .$ 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 $f (x) = \sqrt[]{x} .$

The cube root function is $f (x) = \sqrt[3]{x} .$

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 8.68

For the function $f (x) = \sqrt{2 x - 1},$ find ⓐ $f (5)$ ⓑ $f (−2) .$

Solution

ⓐ

	$f (x) = \sqrt{2 x - 1}$
To evaluate $f (5),$ substitute 5 for $x .$	$f (5) = \sqrt{2 \cdot 5 - 1}$
Simplify.	$f (5) = \sqrt{9}$
Take the square root.	$f (5) = 3$

Step-by-step evaluation of the function f(x) = sqrt(2x - 1) for x=5.

ⓑ

	$f (x) = \sqrt{2 x - 1}$
To evaluate $f (−2),$ substitute −2 for $x .$	$f (−2) = \sqrt{2 (−2) - 1}$
Simplify.	$f (−2) = \sqrt{−5}$

Steps to evaluate f(x)=sqrt(2x-1) at x=-2, showing substitution and simplification to sqrt(-5).

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

Try It 8.135

For the function $f (x) = \sqrt{3 x - 2},$ find ⓐ $f (6)$ ⓑ $f (0) .$

Try It 8.136

For the function $g (x) = \sqrt{5 x + 5},$ find ⓐ $g (4)$ ⓑ $g (−3) .$

We follow the same procedure to evaluate cube roots.

Example 8.69

For the function $g (x) = \sqrt[3]{x - 6},$ find ⓐ $g (14)$ ⓑ $g (−2) .$

Solution

ⓐ

	$g (x) = \sqrt[3]{x - 6}$
To evaluate $g (14),$ substitute 14 for $x .$	$g (14) = \sqrt[3]{14 - 6}$
Simplify.	$g (14) = \sqrt[3]{8}$
Take the cube root.	$g (14) = 2$

This table shows the step-by-step evaluation of the function g(x) = ³√(x-6) for x=14, from substitution to the final result.

ⓑ

	$g (x) = \sqrt[3]{x - 6}$
To evaluate $g (−2),$ substitute −2 for $x .$	$g (−2) = \sqrt[3]{−2 - 6}$
Simplify.	$g (−2) = \sqrt[3]{−8}$
Take the cube root.	$g (−2) = −2$

Step-by-step evaluation of the function g(x) = x−63 for x = -2.

Try It 8.137

For the function $g (x) = \sqrt[3]{3 x - 4},$ find ⓐ $g (4)$ ⓑ $g (1) .$

Try It 8.138

For the function $h (x) = \sqrt[3]{5 x - 2},$ find ⓐ $h (2)$ ⓑ $h (−5) .$

The next example has fourth roots.

Example 8.70

For the function $f (x) = \sqrt[4]{5 x - 4},$ find ⓐ $f (4)$ ⓑ $f (−12)$

Solution

ⓐ

	$f (x) = \sqrt[4]{5 x - 4}$
To evaluate $f (4),$ substitute 4 for $x .$	$f (4) = \sqrt[4]{5 \cdot 4 - 4}$
Simplify.	$f (4) = \sqrt[4]{16}$
Take the fourth root.	$f (4) = 2$

Step-by-step evaluation of the function f(x) = (5x-4)^(1/4) at x=4.

ⓑ

	$f (x) = \sqrt[4]{5 x - 4}$
To evaluate $f (−12),$ substitute −12 for $x .$	$f (−12) = \sqrt[4]{5 (−12) - 4}$
Simplify.	$f (−12) = \sqrt[4]{−64}$

Step-by-step evaluation of the function f(x) = ⁴√(5x-4) at x = -12, showing substitution and simplification.

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

Try It 8.139

For the function $f (x) = \sqrt[4]{3 x + 4},$ find ⓐ $f (4)$ ⓑ $f (−1) .$

Try It 8.140

For the function $g (x) = \sqrt[4]{5 x + 1},$ find ⓐ $g (16)$ ⓑ $g (3) .$

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 $\sqrt[n]{a}$

When n is an even number and:

$a \geq 0,$ then $\sqrt[n]{a}$ is a real number.
$a < 0,$ then $\sqrt[n]{a}$ is not a real number.

When n is an odd number, $\sqrt[n]{a}$ 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 8.71

Find the domain of the function, $f (x) = \sqrt{3 x - 4} .$ Write the domain in interval notation.

Solution

Since the function, $f (x) = \sqrt{3 x - 4}$ 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.

	$3 x - 4 \geq 0$
Solve.	$3 x \geq 4$
	$x \geq \frac{4}{3}$

Steps to solve the inequality 3x - 4 >= 0, showing each mathematical transformation.

The domain of $f (x) = \sqrt{3 x - 4}$ is all values $x \geq \frac{4}{3}$ and we write it in interval notation as $[\frac{4}{3}, \infty) .$

Try It 8.141

Find the domain of the function, $f (x) = \sqrt{6 x - 5} .$ Write the domain in interval notation.

Try It 8.142

Find the domain of the function, $f (x) = \sqrt{4 - 5 x} .$ Write the domain in interval notation.

Example 8.72

Find the domain of the function, $g (x) = \sqrt{\frac{6}{x - 1}} .$ Write the domain in interval notation.

Solution

Since the function, $g (x) = \sqrt{\frac{6}{x - 1}}$ 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 $\frac{6}{x - 1}$ 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 $x - 1 > 0$ and solve.

Solve.

\begin{matrix} x - 1 & > & 0 \\ x & > & 1 \end{matrix}

This table demonstrates the process of solving a simple linear inequality, starting with the instruction and showing the step-by-step solution.

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

We solve $x - 1 = 0$ to find the value that must be eliminated from the domain.

Solve.

\begin{matrix} x - 1 & = & 0 \\ x & = & 1 so x \neq 1 in the domain. \end{matrix}

Demonstration of solving an algebraic equation and determining domain restrictions.

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

Try It 8.143

Find the domain of the function, $f (x) = \sqrt{\frac{4}{x + 3}} .$ Write the domain in interval notation.

Try It 8.144

Find the domain of the function, $h (x) = \sqrt{\frac{9}{x - 5}} .$ Write the domain in interval notation.

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

Example 8.73

Find the domain of the function, $f (x) = \sqrt[3]{2 x^{2} + 3} .$ Write the domain in interval notation.

Solution

Since the function, $f (x) = \sqrt[3]{2 x^{2} + 3}$ 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 $(− \infty, \infty) .$

The domain of $f (x) = \sqrt[3]{2 x^{2} + 3}$ is all real numbers and we write it in interval notation as $(− \infty, \infty) .$

Try It 8.145

Find the domain of the function, $f (x) = \sqrt[3]{3 x^{2} - 1} .$ Write the domain in interval notation.

Try It 8.146

Find the domain of the function, $g (x) = \sqrt[3]{5 x - 4} .$ Write the domain in interval notation.

Graph Radical Functions

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

This tells us the domain is $x \geq 0$ and we write this in interval notation as $[0, \infty) .$

Previously we used point plotting to graph the function, $f (x) = \sqrt{x} .$ 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).

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 $[0, \infty) .$

Example 8.74

For the function $f (x) = \sqrt{x + 3},$

ⓐ find the domain ⓑ graph the function ⓒ use the graph to determine the range.

Solution

ⓐ Since the radical has index 2, we know the radicand must be greater than or equal to zero. If $x + 3 \geq 0,$ then $x \geq −3 .$ This tells us the domain is all values $x \geq −3$ and written in interval notation as $[−3, \infty) .$

ⓑ To graph the function, we choose points in the interval $[−3, \infty)$ 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).

Try It 8.147

For the function $f (x) = \sqrt{x + 2},$ ⓐ find the domain ⓑ graph the function ⓒ use the graph to determine the range.

Try It 8.148

For the function $f (x) = \sqrt{x - 2},$ ⓐ find the domain ⓑ graph the function ⓒ use the graph to determine the range.

In our previous work graphing functions, we graphed $f (x) = x^{3}$ but we did not graph the function $f (x) = \sqrt[3]{x} .$ We will do this now in the next example.

Example 8.75

For the function $f (x) = \sqrt[3]{x},$ ⓐ find the domain ⓑ graph the function ⓒ use the graph to determine the range.

Solution

ⓐ 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 $(− \infty, \infty)$

ⓑ To graph the function, we choose points in the interval $(− \infty, \infty)$ 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).

Try It 8.149

For the function $f (x) = − \sqrt[3]{x},$

ⓐ find the domain ⓑ graph the function ⓒ use the graph to determine the range.

Try It 8.150

For the function $f (x) = \sqrt[3]{x - 2},$

ⓐ find the domain ⓑ graph the function ⓒ use the graph to determine the range.

Media

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

Section 8.7 Exercises

Practice Makes Perfect

Evaluate a Radical Function

In the following exercises, evaluate each function.

351.

$f (x) = \sqrt{4 x - 4},$ find ⓐ $f (5)$ ⓑ $f (0) .$

352.

$f (x) = \sqrt{6 x - 5},$ find ⓐ $f (5)$ ⓑ $f (−1) .$

353.

$g (x) = \sqrt{6 x + 1},$ find ⓐ $g (4)$ ⓑ $g (8) .$

354.

$g (x) = \sqrt{3 x + 1},$ find ⓐ $g (8)$ ⓑ $g (5) .$

355.

$F (x) = \sqrt{3 - 2 x},$ find ⓐ $F (1)$ ⓑ $F (−11) .$

356.

$F (x) = \sqrt{8 - 4 x},$ find ⓐ $F (1)$ ⓑ $F (−2) .$

357.

$G (x) = \sqrt{5 x - 1},$ find ⓐ $G (5)$ ⓑ $G (2) .$

358.

$G (x) = \sqrt{4 x + 1},$ find ⓐ $G (11)$ ⓑ $G (2) .$

359.

$g (x) = \sqrt[3]{2 x - 4},$ find ⓐ $g (6)$ ⓑ $g (−2) .$

360.

$g (x) = \sqrt[3]{7 x - 1},$ find ⓐ $g (4)$ ⓑ $g (−1) .$

361.

$h (x) = \sqrt[3]{x^{2} - 4},$ find ⓐ $h (−2)$ ⓑ $h (6) .$

362.

$h (x) = \sqrt[3]{x^{2} + 4},$ find ⓐ $h (−2)$ ⓑ $h (6) .$

363.

For the function $f (x) = \sqrt[4]{2 x^{3}},$ find ⓐ $f (0)$ ⓑ $f (2) .$

364.

For the function $f (x) = \sqrt[4]{3 x^{3}},$ find ⓐ $f (0)$ ⓑ $f (3) .$

365.

For the function $g (x) = \sqrt[4]{4 - 4 x},$ find ⓐ $g (1)$ ⓑ $g (−3) .$

366.

For the function $g (x) = \sqrt[4]{8 - 4 x},$ find ⓐ $g (−6)$ ⓑ $g (2) .$

Find the Domain of a Radical Function

In the following exercises, find the domain of the function and write the domain in interval notation.

367.

$f (x) = \sqrt{3 x - 1}$

368.

$f (x) = \sqrt{4 x - 2}$

369.

$g (x) = \sqrt{2 - 3 x}$

370.

$g (x) = \sqrt{8 - x}$

371.

$h (x) = \sqrt{\frac{5}{x - 2}}$

372.

$h (x) = \sqrt{\frac{6}{x + 3}}$

373.

$f (x) = \sqrt{\frac{x + 3}{x - 2}}$

374.

$f (x) = \sqrt{\frac{x - 1}{x + 4}}$

375.

$g (x) = \sqrt[3]{8 x - 1}$

376.

$g (x) = \sqrt[3]{6 x + 5}$

377.

$f (x) = \sqrt[3]{4 x^{2} - 16}$

378.

$f (x) = \sqrt[3]{6 x^{2} - 25}$

379.

$F (x) = \sqrt[4]{8 x + 3}$

380.

$F (x) = \sqrt[4]{10 - 7 x}$

381.

$G (x) = \sqrt[5]{2 x - 1}$

382.

$G (x) = \sqrt[5]{6 x - 3}$

Graph Radical Functions

In the following exercises, ⓐ find the domain of the function ⓑ graph the function ⓒ use the graph to determine the range.

383.

$f (x) = \sqrt{x + 1}$

384.

$f (x) = \sqrt{x - 1}$

385.

$g (x) = \sqrt{x + 4}$

386.

$g (x) = \sqrt{x - 4}$

387.

$f (x) = \sqrt{x} + 2$

388.

$f (x) = \sqrt{x} - 2$

389.

$g (x) = 2 \sqrt{x}$

390.

$g (x) = 3 \sqrt{x}$

391.

$f (x) = \sqrt{3 - x}$

392.

$f (x) = \sqrt{4 - x}$

393.

$g (x) = − \sqrt{x}$

394.

$g (x) = − \sqrt{x} + 1$

395.

$f (x) = \sqrt[3]{x + 1}$

396.

$f (x) = \sqrt[3]{x - 1}$

397.

$g (x) = \sqrt[3]{x + 2}$

398.

$g (x) = \sqrt[3]{x - 2}$

399.

$f (x) = \sqrt[3]{x} + 3$

400.

$f (x) = \sqrt[3]{x} - 3$

401.

$g (x) = \sqrt[3]{x}$

402.

$g (x) = − \sqrt[3]{x}$

403.

$f (x) = 2 \sqrt[3]{x}$

404.

$f (x) = −2 \sqrt[3]{x}$

Writing Exercises

405.

Explain how to find the domain of a fourth root function.

406.

Explain how to find the domain of a fifth root function.

407.

Explain why $y = \sqrt[3]{x}$ is a function.

408.

Explain why the process of finding the domain of a radical function with an even index is different from the process when the index is odd.

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

The table has 4 columns and 4 rows. The first row is a header row with the headers “I can…”, “Confidently”, “With some help.”, and “No – I don’t get it!”. The first column contains the phrases “evaluate a radical function”, “find the domain of a radical function”, and “graph a radical function”. The other columns are left blank so the learner can indicate their level of understanding.

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?

8.7 Use Radicals in Functions

Learning Objectives

Be Prepared 8.19

Be Prepared 8.20

Be Prepared 8.21

Evaluate a Radical Function

Radical Function

Example 8.68

Solution

Try It 8.135

Try It 8.136

Example 8.69

Solution

Try It 8.137

Try It 8.138

Example 8.70

Solution

Try It 8.139

Try It 8.140

Find the Domain of a Radical Function

Properties of a n a n

Domain of a Radical Function

Example 8.71

Solution

Try It 8.141

Try It 8.142

Example 8.72

Solution

Try It 8.143

Try It 8.144

Example 8.73

Solution

Try It 8.145

Try It 8.146

Graph Radical Functions

Example 8.74

Solution

Try It 8.147

Try It 8.148

Example 8.75

Solution

Try It 8.149

Try It 8.150

Media

Section 8.7 Exercises

Practice Makes Perfect

Writing Exercises

Self Check

Properties of $\sqrt[n]{a}$