# K12MATH011: Algebra II

Radical functions contain expressions under radical (nth root) signs. For instance f(x) = 2x+5 is an example of a radical function. Solving them requires raising both sides of an equation to a power, which may create extraneous roots (solutions). These potential solutions have to be checked, in order to make sure one or more make the equation true. As with rational functions, the methods for solving radical equations arise from what you learned from polynomials. As with rational functions, the answers found will require a check, as some potential solutions may cause untrue statements.

Completing this unit should take approximately 10 hours and 45 minutes.

☐    Subunit 6.1: 1 hour and 45 minutes

☐    Subunit 6.2: 1 hour and 30 minutes

☐    Subunit 6.3: 1 hour and 45 minutes

☐    Subunit 6.4: 1 hour and 30 minutes

☐    Subunit 6.5: 1 hour and 30 minutes

☐    Subunit 6.6: 2 hours and 45 minutes

Unit6 Learning Outcomes
Upon successful completion of this unit, you will be able to: - Rewrite a radical expression with rational exponents and vice versa. - Determine the results of operations on radical functions. - Determine the results of composites of radical functions. - Find the inverse of a radical function, if one exists. - Graph a radical function. - Find solutions to radical equations and inequalities.

6.1 The Connection between nth Roots and Rational Exponents   The connection between nth roots and rational exponents is that na = a1/n  by definition. Exponent rules make simplifying nth root expressions easy. There are applications of radical equations in physics, physical chemistry, biology, and business.For example, in biology, functions for determining population growth, given certain factors, may be expressed as a radical equation. In physics, equations of actions on a subatomic scale may be expressed using radicals.

• Explanation: CK-12: “Exponents” Link: CK-12: “Exponents” (HTML)

Instructions: This page contains practice problems and examples on exponents, including fractional exponents, which are the same as nth roots. Work all the problems in the “Fractional Exponents” section. That is, 5x = x1/5. Find out how this definition can help solve rational equations. Work all the problems in the “Fractional Exponents” section.

Completing this activity should take approximately 1 hour.

• Explanation: GeoGebra: “Fractional Exponent Practice” Link: GeoGebra: “Fractional Exponent Practice” (HTML)

Instructions: This interactive application allows you to see radical equations solved. This page does a great job showing the connection between radicals and fractional exponents.

Practicing with this application and writing notes should take approximately 45 minutes.

6.2 Operations on Radical Functions   As with any set of functions, you may add, subtract, multiply, and divide radical functions. Take care once an operation is performed, however. The domain and range of the result may be different from the original functions.

Instructions: This is an interactive self-quiz that allows you to add and subtract radical expressions. Note that there are hints available if needed.

Completing the practice problems and writing notes should take approximately 45 minutes.

Instructions: This is an interactive self-quiz that allows you to multiply and simplify radical expressions. Note that there are hints available if needed.

Completing the practice problems and writing notes should take approximately 45 minutes.

6.3 Compositions of Radical Functions   As with any function, radicals can form compositions. That is, functions can be evaluated for other functions as well as numerical values (i.e., we can find f(2x-1) as easily as we can find f(3)). Caution has to be taken, however, as the domain and range of the composition may be very different from those of the original functions that make up the composition. In other words, always check any solutions found to make sure they do not cause a problem in the equation, as in a negative under a square root or a zero in a denominator.

• Explanation: Math Planet: “Composition of Functions” Link: Math Planet: “Composition of Functions” (YouTube)

Instructions: This video explains how to perform general composition on functions. Try working though five or six problems. Make sure to check the domain at the end of your work, as square root functions have domains limited to values that yield zero or positive values under the radical sign for real numbers.

Watching the video and writing notes should take approximately 15 minutes.

• Explanation: CK-12: “Composition of Functions” Link: CK-12: “Composition of Functions” (HTML and YouTube)

Instructions: Read and work through this entire page, which begins with a description, then a video, a guidance section, vocabulary, and practice problems explaining how to perform general composition on functions. Make sure you check the domain at the end of each practice problem, as square root functions have domains limited to values that yield zero or positive values under the radical sign for real numbers.

Completing this activity should take approximately 1 hour and 30 minutes.

6.4 Inverses   Inverse functions are one-to-one (i.e., each x is matched to a unique y and vice versa) and onto (i.e., every possible x matches to every possible y). In general, if (a,b) is an element of a one-to-one function, (b,a) will be an element of the inverse.

• Explanation: Carl Stitz and Jeff Zeager’s Precalculus: “Chapter 5, Section 2: Inverse Functions” Link: Carl Stitz and Jeff Zeager’s Precalculus: “Chapter 5, Section 2: Inverse Functions” (PDF)

Instructions: This chapter is devoted to a detailed discussion of inverse functions, including conditions necessary to determine if a function has a proper inverse. Radical functions and their inverses are included. Read pages 378-393, and then try the problems at the end of the section. Answers are on the pages that follow.

Completing this activity should take approximately 1 hour and 30 minutes.

6.5 Graphs of Radical Functions   Graphs of radical functions are generally easy to draw. The basic function f(x) = nx  usually starts at (0,0) and has (1,1) as an element. If n is odd, (-1,-1) it is also an element. The graph gets flatter the larger n is, but all such functions have a basic smooth curve.

• Explanation: Carl Stitz and Jeff Zeager’s Precalculus: “Chapter 5: Further Topics in Functions” Link: Carl Stitz and Jeff Zeager’s Precalculus: “Chapter 5: Further Topics in Functions” (PDF)

Instructions: This is a detailed discussion of radical functions, including their graphs. Make sure to check the domain of the functions in every example.

Reading the material and writing notes should take approximately 45 minutes.

Instructions: This interactive graph allows you to change values in a transformation form of a radical function to see how the graph changes. There are three sliders: One is for the coefficient in front of the radical, one is for the horizontal shift value, and one is for the vertical shift value. Change the values to see how the graph changes.

Practicing with this application and writing notes should take approximately 45 minutes.

6.6 Solving Radical Equations and Inequalities   Radical equations are solvable by raising both sides of the equation to the power of the radical. That is, to eliminate a cube root, raise both sides of the equation to the third power. The problem is that any solution found must be checked to ensure no negatives are caused under even roots. Such answers are complex, and the equations we will be solving should have real numbers solutions.

Instructions: This video discusses the process for solving radical equations. View it as many times as necessary to gain understanding.

Watching the video and writing notes should take approximately 15 minutes.

• Explanation: CK-12: “Equations with Radicals on Both Sides” Link: CK-12: “Equations with Radicals on Both Sides” (Vimeo and HTML)

Instructions: This page begins with a video discussing the process for solving radical equations. View as many times as necessary to gain understanding. The page also contains detailed discussion on solving radical equations with radicals on both sides of the equation. There are examples and practice problems at the end. In the middle of the webpage, there is a section entitled “Identify Extraneous Solutions to Radical Equations.” Read this section carefully, and then write a one-page essay on how extraneous solutions occur and why it is necessary to always check your answer.

Completing this activity should take approximately 1 hour and 30 minutes.