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Intermediate Algebra

8.7 Use Radicals in Functions

Intermediate Algebra8.7 Use Radicals in Functions
  1. Preface
  2. 1 Foundations
    1. Introduction
    2. 1.1 Use the Language of Algebra
    3. 1.2 Integers
    4. 1.3 Fractions
    5. 1.4 Decimals
    6. 1.5 Properties of Real Numbers
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 Solving Linear Equations
    1. Introduction
    2. 2.1 Use a General Strategy to Solve Linear Equations
    3. 2.2 Use a Problem Solving Strategy
    4. 2.3 Solve a Formula for a Specific Variable
    5. 2.4 Solve Mixture and Uniform Motion Applications
    6. 2.5 Solve Linear Inequalities
    7. 2.6 Solve Compound Inequalities
    8. 2.7 Solve Absolute Value Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Graphs and Functions
    1. Introduction
    2. 3.1 Graph Linear Equations in Two Variables
    3. 3.2 Slope of a Line
    4. 3.3 Find the Equation of a Line
    5. 3.4 Graph Linear Inequalities in Two Variables
    6. 3.5 Relations and Functions
    7. 3.6 Graphs of Functions
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Systems of Linear Equations
    1. Introduction
    2. 4.1 Solve Systems of Linear Equations with Two Variables
    3. 4.2 Solve Applications with Systems of Equations
    4. 4.3 Solve Mixture Applications with Systems of Equations
    5. 4.4 Solve Systems of Equations with Three Variables
    6. 4.5 Solve Systems of Equations Using Matrices
    7. 4.6 Solve Systems of Equations Using Determinants
    8. 4.7 Graphing Systems of Linear Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Polynomials and Polynomial Functions
    1. Introduction
    2. 5.1 Add and Subtract Polynomials
    3. 5.2 Properties of Exponents and Scientific Notation
    4. 5.3 Multiply Polynomials
    5. 5.4 Dividing Polynomials
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Factoring
    1. Introduction to Factoring
    2. 6.1 Greatest Common Factor and Factor by Grouping
    3. 6.2 Factor Trinomials
    4. 6.3 Factor Special Products
    5. 6.4 General Strategy for Factoring Polynomials
    6. 6.5 Polynomial Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 Rational Expressions and Functions
    1. Introduction
    2. 7.1 Multiply and Divide Rational Expressions
    3. 7.2 Add and Subtract Rational Expressions
    4. 7.3 Simplify Complex Rational Expressions
    5. 7.4 Solve Rational Equations
    6. 7.5 Solve Applications with Rational Equations
    7. 7.6 Solve Rational Inequalities
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Roots and Radicals
    1. Introduction
    2. 8.1 Simplify Expressions with Roots
    3. 8.2 Simplify Radical Expressions
    4. 8.3 Simplify Rational Exponents
    5. 8.4 Add, Subtract, and Multiply Radical Expressions
    6. 8.5 Divide Radical Expressions
    7. 8.6 Solve Radical Equations
    8. 8.7 Use Radicals in Functions
    9. 8.8 Use the Complex Number System
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Quadratic Equations and Functions
    1. Introduction
    2. 9.1 Solve Quadratic Equations Using the Square Root Property
    3. 9.2 Solve Quadratic Equations by Completing the Square
    4. 9.3 Solve Quadratic Equations Using the Quadratic Formula
    5. 9.4 Solve Quadratic Equations in Quadratic Form
    6. 9.5 Solve Applications of Quadratic Equations
    7. 9.6 Graph Quadratic Functions Using Properties
    8. 9.7 Graph Quadratic Functions Using Transformations
    9. 9.8 Solve Quadratic Inequalities
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Exponential and Logarithmic Functions
    1. Introduction
    2. 10.1 Finding Composite and Inverse Functions
    3. 10.2 Evaluate and Graph Exponential Functions
    4. 10.3 Evaluate and Graph Logarithmic Functions
    5. 10.4 Use the Properties of Logarithms
    6. 10.5 Solve Exponential and Logarithmic Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  12. 11 Conics
    1. Introduction
    2. 11.1 Distance and Midpoint Formulas; Circles
    3. 11.2 Parabolas
    4. 11.3 Ellipses
    5. 11.4 Hyperbolas
    6. 11.5 Solve Systems of Nonlinear Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  13. 12 Sequences, Series and Binomial Theorem
    1. Introduction
    2. 12.1 Sequences
    3. 12.2 Arithmetic Sequences
    4. 12.3 Geometric Sequences and Series
    5. 12.4 Binomial Theorem
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  14. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
  15. Index

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 8.7

Before you get started, take this readiness quiz.

  1. Solve: 12x0.12x0.
    If you missed this problem, review Example 2.50.
  2. For f(x)=3x4,f(x)=3x4, evaluate f(2),f(−1),f(0).f(2),f(−1),f(0).
    If you missed this problem, review Example 3.48.
  3. Graph f(x)=x.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.f(x)=x.

The cube root function is f(x)=x3.f(x)=x3.

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)=2x1,f(x)=2x1, find f(5)f(5) f(−2).f(−2).

Try It 8.135

For the function f(x)=3x2,f(x)=3x2, find f(6)f(6) f(0).f(0).

Try It 8.136

For the function g(x)=5x+5,g(x)=5x+5, find g(4)g(4) g(−3).g(−3).

We follow the same procedure to evaluate cube roots.

Example 8.69

For the function g(x)=x63,g(x)=x63, find g(14)g(14) g(−2).g(−2).

Try It 8.137

For the function g(x)=3x43,g(x)=3x43, find g(4)g(4) g(1).g(1).

Try It 8.138

For the function h(x)=5x23,h(x)=5x23, find h(2)h(2) h(−5).h(−5).

The next example has fourth roots.

Example 8.70

For the function f(x)=5x44,f(x)=5x44, find f(4)f(4) f(−12)f(−12)

Try It 8.139

For the function f(x)=3x+44,f(x)=3x+44, find f(4)f(4) f(−1).f(−1).

Try It 8.140

For the function g(x)=5x+14,g(x)=5x+14, find g(16)g(16) g(3).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 anan

When n is an even number and:

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

When n is an odd number, anan 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)=3x4.f(x)=3x4. Write the domain in interval notation.

Try It 8.141

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

Try It 8.142

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

Example 8.72

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

Try It 8.143

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

Try It 8.144

Find the domain of the function, h(x)=9x5.h(x)=9x5. 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)=2x2+33.f(x)=2x2+33. Write the domain in interval notation.

Try It 8.145

Find the domain of the function, f(x)=3x213.f(x)=3x213. Write the domain in interval notation.

Try It 8.146

Find the domain of the function, g(x)=5x43.g(x)=5x43. 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)=x,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 x0x0 and we write this in interval notation as [0,).[0,).

Previously we used point plotting to graph the function, f(x)=x.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).

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,).[0,).

Example 8.74

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

find the domain graph the function use the graph to determine the range.

Try It 8.147

For the function f(x)=x+2,f(x)=x+2, find the domain graph the function use the graph to determine the range.

Try It 8.148

For the function f(x)=x2,f(x)=x2, find the domain graph the function use the graph to determine the range.

In our previous work graphing functions, we graphed f(x)=x3f(x)=x3 but we did not graph the function f(x)=x3.f(x)=x3. We will do this now in the next example.

Example 8.75

For the function f(x)=x3,f(x)=x3, find the domain graph the function use the graph to determine the range.

Try It 8.149

For the function f(x)=x3,f(x)=x3,

find the domain graph the function use the graph to determine the range.

Try It 8.150

For the function f(x)=x23,f(x)=x23,

find the domain graph the function use the graph to determine the range.

Media Access Additional Online Resources

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)=4x4,f(x)=4x4, find f(5)f(5) f(0).f(0).

352.

f(x)=6x5,f(x)=6x5, find f(5)f(5) f(−1).f(−1).

353.

g(x)=6x+1,g(x)=6x+1, find g(4)g(4) g(8).g(8).

354.

g(x)=3x+1,g(x)=3x+1, find g(8)g(8) g(5).g(5).

355.

F(x)=32x,F(x)=32x, find F(1)F(1) F(−11).F(−11).

356.

F(x)=84x,F(x)=84x, find F(1)F(1) F(−2).F(−2).

357.

G(x)=5x1,G(x)=5x1, find G(5)G(5) G(2).G(2).

358.

G(x)=4x+1,G(x)=4x+1, find G(11)G(11) G(2).G(2).

359.

g(x)=2x43,g(x)=2x43, find g(6)g(6) g(−2).g(−2).

360.

g(x)=7x13,g(x)=7x13, find g(4)g(4) g(−1).g(−1).

361.

h(x)=x243,h(x)=x243, find h(−2)h(−2) h(6).h(6).

362.

h(x)=x2+43,h(x)=x2+43, find h(−2)h(−2) h(6).h(6).

363.

For the function f(x)=2x34,f(x)=2x34, find f(0)f(0) f(2).f(2).

364.

For the function f(x)=3x34,f(x)=3x34, find f(0)f(0) f(3).f(3).

365.

For the function g(x)=44x4,g(x)=44x4, find g(1)g(1) g(−3).g(−3).

366.

For the function g(x)=84x4,g(x)=84x4, find g(−6)g(−6) g(2).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)=3x1f(x)=3x1

368.

f(x)=4x2f(x)=4x2

369.

g(x)=23xg(x)=23x

370.

g(x)=8xg(x)=8x

371.

h(x)=5x2h(x)=5x2

372.

h(x)=6x+3h(x)=6x+3

373.

f(x)=x+3x2f(x)=x+3x2

374.

f(x)=x1x+4f(x)=x1x+4

375.

g(x)=8x13g(x)=8x13

376.

g(x)=6x+53g(x)=6x+53

377.

f(x)=4x2163f(x)=4x2163

378.

f(x)=6x2253f(x)=6x2253

379.

F(x)=8x+34F(x)=8x+34

380.

F(x)=107x4F(x)=107x4

381.

G(x)=2x15G(x)=2x15

382.

G(x)=6x35G(x)=6x35

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)=x+1f(x)=x+1

384.

f(x)=x1f(x)=x1

385.

g(x)=x+4g(x)=x+4

386.

g(x)=x4g(x)=x4

387.

f(x)=x+2f(x)=x+2

388.

f(x)=x2f(x)=x2

389.

g(x)=2xg(x)=2x

390.

g(x)=3xg(x)=3x

391.

f(x)=3xf(x)=3x

392.

f(x)=4xf(x)=4x

393.

g(x)=xg(x)=x

394.

g(x)=x+1g(x)=x+1

395.

f(x)=x+13f(x)=x+13

396.

f(x)=x13f(x)=x13

397.

g(x)=x+23g(x)=x+23

398.

g(x)=x23g(x)=x23

399.

f(x)=x3+3f(x)=x3+3

400.

f(x)=x33f(x)=x33

401.

g(x)=x3g(x)=x3

402.

g(x)=x3g(x)=x3

403.

f(x)=2x3f(x)=2x3

404.

f(x)=−2x3f(x)=−2x3

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=x3y=x3 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?

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