Unit 7 Inverse and Radical Functions

Test date: _____________________

Name: ___________________________________________________________________________________

By the end of this unit, you will be able to…

Find the inverse of a function and identify its domain and range Compose two functions and identify the domain and range Evaluate the inverse of a function in multiple representations Determine if functions are inverses using compositions Find the domain and range of a radical function Graph a radical function or inequality and identify its zeros from its x-intercepts Recognize shifts of radical functions and apply them to a graph Simplify expressions with rational exponents and nth roots Rationalize denominators Solve equations involving radicals and rational exponents Identify extraneous solutions Solve radical inequalities in one variable

2

Table of Contents

6.1 Operations on Functions ......................................................................................................................................................... 4

Function Composition................................................................................................................................................................. 5

6.2 Finding Inverse Functions ...................................................................................................................................................... 6

Proving Inverse Functions ........................................................................................................................................................ 7

Finding Inverses from Tables and Graphs .......................................................................................................................... 8

6.3 Graphing Square Root and Cube Root Functions .......................................................................................................... 8

Radical Inequalities ................................................................................................................................................................... 11

Applications: ................................................................................................................................................................................. 13

6.4 nth Roots ..................................................................................................................................................................................... 14

Simplifying Radicals .................................................................................................................................................................. 14

Approximate Radicals with a Calculator ........................................................................................................................... 15

Solving Equations using nth Roots ...................................................................................................................................... 15

Applications .................................................................................................................................................................................. 16

6.5 Operations with Radical Expressions .............................................................................................................................. 17

The Product Property of Radicals ........................................................................................................................................ 17

Quotient Property of Radicals ............................................................................................................................................... 18

Operations on Radicals ............................................................................................................................................................. 18

Rationalizing Denominators................................................................................................................................................... 20

Basic Rational Exponents ........................................................................................................................................................ 20

6.6 Rational Exponents ................................................................................................................................................................. 21

Simplifying Radicals with Different Indices ..................................................................................................................... 22

Using Conjugates to Rationalize the Denominator ........................................................................................................ 23

6.7 Solving Radical Equations and Inequalities ................................................................................................................... 24

Solving Equations with Rational Exponents .................................................................................................................... 25

Solving Inequalities.................................................................................................................................................................... 26

3

Where are the functions farthest apart? Find the maximum vertical distance d between the parabola and the line for the shaded region.

4

6.1 Operations on Functions

1. What is the maximum vertical distance between these two curves?

2. In a particular county, the population of the two largest cities can be modeled by 𝑓(𝑥) = 200𝑥 +

25 and 𝑔(𝑥) = 175𝑥 − 15, where x is the number of years since 2000 and the population is in thousands.

a. What is the population of the two cities combined after 10 years? After any number of years?

b. What is the difference in the populations of the two cities after 10 years? After any number of years?

Operation Definition Example:

f(x) = 2x, and g(x) = -x + 5

Sum (f + g)(x) = __________________

(f + g)(x) = __________________________________________

Difference (f – g)(x) = _________________

(f – g)(x) = __________________________________________

Product (f ⋅ g)(x) = __________________

(f ⋅ g)(x) = ___________________________________________

Quotient (𝑓

𝑔)(x) = __________

(𝑓

𝑔)(x) = ___________________________________________

𝑓(𝑥) = 2𝑥 − 1

𝑔(𝑥) = 𝑥3

5

Function Composition

In a composition of functions, the _____________________________________________________________

_________________________________________________________________________________________.

[ ]( )f g x _________________ domain of g range of g, domain of f range of f

__________ ________ ________

The composition of functions may ______________________. Given two functions f and g,

[ ]( )f g x is defined only if _______________________________________________________________________________.

[ ]( )g f x is defined only if _______________________________________________________________________________.

Example: For f(x) = 4x2 – 2, and g(x) = 𝑥

2, find the following:

a.) Find f[g(6)] b.) Find f[f(-1)]

Examples: Find [g ◦ h](x), [h ◦ g](x), and [g ◦ g](x) for g(x) = 3x – 4 and h(x) = 𝑥2 – 1. Identify the domain and range.

[g ◦ h](x) [h ◦ g](x) [g ◦ g](x)

Examples: For each pair of functions, find 𝑓 ∘ 𝑔 and 𝑔 ∘ 𝑓, if they exist.

1.

2.

[ ]( )f g x

6

6.2 Finding Inverse Functions 1. Use a table of values to graph f(x) = 2x – 4 2. Switch the x and y values from #1, plot the new

points and draw the line. Write the equation of a line that passes through those points (use y = mx + b).

Eq:______________________ 3. In words, f(x) takes an x-value, multiplies it by 2 and then subtracts 4. Describe the equation from #2 in words: ______________________________________________________________

4. Repeat this process once more for f(x) = 𝑥−3

2 5. Switch the x and y values from #4, plot the new

points and write the equation of a line that passes through those points (use y = mx + b)

Eq:___________________

6. In words f(x) subtracts 3 from x, then divides the result by 2. Describe the equation from 5 in words: _______________________________________________________________ 7. Notice how this description “inverts” the description of f(x). How could you rewrite your description from #3 so that it better “inverts” the description of f(x) but is still accurate?

x y

-2

-1

0

1

2

x y

x y

-3

-1

0

1

3

x y

http://www.google.com/url?sa=i&source=images&cd=&cad=rja&docid=rvDEFF-nCkg-CM&tbnid=Pl3aj5jpeSEEGM:&ved=0CAgQjRwwAA&url=http://www.algebra-class.com/graphing-equations.html&ei=r8s-Ur2sHo2wqQGhwYDgCg&psig=AFQjCNHtbL9hgk0RiS3qw1PxNef1hmn-JQ&ust=1379933487549291http://www.google.com/url?sa=i&source=images&cd=&cad=rja&docid=rvDEFF-nCkg-CM&tbnid=Pl3aj5jpeSEEGM:&ved=0CAgQjRwwAA&url=http://www.algebra-class.com/graphing-equations.html&ei=r8s-Ur2sHo2wqQGhwYDgCg&psig=AFQjCNHtbL9hgk0RiS3qw1PxNef1hmn-JQ&ust=1379933487549291http://www.google.com/url?sa=i&source=images&cd=&cad=rja&docid=rvDEFF-nCkg-CM&tbnid=Pl3aj5jpeSEEGM:&ved=0CAgQjRwwAA&url=http://www.algebra-class.com/graphing-equations.html&ei=r8s-Ur2sHo2wqQGhwYDgCg&psig=AFQjCNHtbL9hgk0RiS3qw1PxNef1hmn-JQ&ust=1379933487549291http://www.google.com/url?sa=i&source=images&cd=&cad=rja&docid=rvDEFF-nCkg-CM&tbnid=Pl3aj5jpeSEEGM:&ved=0CAgQjRwwAA&url=http://www.algebra-class.com/graphing-equations.html&ei=r8s-Ur2sHo2wqQGhwYDgCg&psig=AFQjCNHtbL9hgk0RiS3qw1PxNef1hmn-JQ&ust=1379933487549291

7

In general an inverse function maps the output values of a function back to their original input values. In order to find an inverse function, simply switch _____ and _____, and ____________________________________. The resulting function is its inverse, f-1(x). Exercises: Find the inverse f-1(x) of each:

8. f(x) = 2x – 3 9. f(x) = ½x3 + 4 10. f(x) = 2

3𝑥+5

11. f(x) = 𝑥2+2

5 12. f(x) = (2x + 4)2 – 3 13. 𝑓(𝑥) = √𝑥 − 1

Proving Inverse Functions

14. For f(x) = 2x – 4 and g(x) = 𝑥+4

2, find:

a) f(g(x)) = b) g(f(x)) =

15. For f(x) = 𝑥2+2

5 and g(x) = √5𝑥 − 2 , find:

a) f(g(x)) = b) g(f(x)) = What do you notice about all of the solutions to all of the compositions of functions that you found? __________________________________________________________________________________________ If two functions f(x) and g(x) are inverses, f(g(x)) = ______ and g(f(x))=_______. If for all values of x, f(g(x)) = x and g(f(x))=x, then the two functions f(x) and g(x) are ___________________. We can use this to prove that two functions are or are not inverses.

8

Finding Inverses from Tables and Graphs 16. The table below shows several values for the function f(x). If f(x) is a one-to-one function, find each of the values. a. f-1(-3)=_______ b. f-1(0)=_______ c. f-1(-4)=_______ d. f-1(1) =_______ 17. The graph of the function g(x) is shown in the coordinate plane to the right. Find each of the values. Approximate if needed. a. f-1(3)=_______ b. f-1(-1)=_______ c. f-1(-5)=_______ d. f-1(-9) =_______

x f(x) -4 1 -3 0.5 -2 0 -1 -2 0 -3 1 -3.5 2 -4

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6.3 Graphing Square Root and Cube Root Functions 1. Use your calculator to graph the following radical functions below:

a) y x b) y x 3

Domain:_________ Range:___________ Domain:_____________ Range:____________ 3. Graph the following functions on the same planes above, and find the points as you did above.

a) xy 2 b) y x 23

4. Explain what the coefficient 2 did to each graph: 5. Did the domain and range of either graph change? 6. Now, determine what the coefficient of ½ will do to each graph.

a) y x1

2 b) y x

1

2

3

x y

-1

0

1

4

9

x Y

-8

-1

0

1

8

X y

-8

-1

0

1

8

x y

0

1

4

9

10

7. Graph the following functions, what effect does the negative sign have on the original graphs?

a) y x b) y x 3

8. What are the domain and Range of (a) and (b) a) b)

7. Graph a) y x 2 and b) y x 23 , what effect does adding 2 have on the original graphs of y x

and y x 3 ? (what direction did the graphs move?)

a) b)

Domain:________________________ Domain:______________________ Range:_________________________ Range:_______________________

8. Graph y x 2 and y x 23 , what effect does subtracting 2 have on the original graphs of y x

and y x 3 ? (what direction did the graphs move?)

D: D: R: R:

x y

-8

-1

0

1

8

x y

0

1

4

9

x y

x y

11

9. Graph y x 2 and y x 2 , what effect does this have on the original graphs? (what direction did

the graphs move?) D: D: R: R: General Rules: How do a, h, and k affect the original graphs?

y a x h k y a x h k 3

a:_______________________________ k:_____________________________

h:_______________________________

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Radical Inequalities

1. Graph 𝑦 < √𝑥 − 4 − 6

2. Graph 𝑓(𝑥) ≥ √2𝑥 + 1

3. Graph 𝑓(𝑥) < −√𝑥3

+ 4

“On my way to teach those kids some RADICAL functions!”

13

Applications: 1. The pitch of a certain string on an instrument (measure in hertz) can be modeled by the function:

f(T) = 1

1.28√

𝑇

.0000708, where T is the tension of the string in Kilograms.

a) Make a table of values for the T values 1-10.

b) How much tension will be needed for a pitch of 200 Hz?

c) What is the domain of this function?

2. The speed of a tsunami can be modeled by the equation v = 356√𝑑, where v is the speed in kilometers per hour, and d is the average depth of the water. If a tsunami is found to be traveling at 145 kilometers per hour, what is the average depth of the water?

3. The approximate time t (in seconds) that it takes an object to fall a distance of d feet is given by t = √𝑑

16.

Suppose a parachutist falls 11 seconds before the parachute opens, how far does the parachutist fall during that time? What is the domain of this function?

4. The velocity of a roller coaster as it moves down a hill is V = √𝑣2 + 64ℎ, where v is the initial velocity in feet per second and h is the vertical drop in feet. The coaster designer wants the coaster to have a velocity of 90 feet per second. If the initial velocity of the coaster at the top of the hill is 10 feet per second, how high should they make the hill?

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6.4 nth Roots

Simplifying Radicals

Square Root For any real numbers a and b, _________________, then a is a square root of b.

nth Root For any real numbers a and b, and any positive integer n, _______, then a is an nth root of b.

Real nth Roots of b,

√𝒃𝒏

, – √𝒃𝒏

1. If n is even and b > 0, then b has _____________________________________________

2. If n is odd and b > 0, then b has ______________________________________________

3. If n is even and b < 0, then __________________________________________________

4. If n is odd and b < 0, then ___________________________________________________

Simplify.

Example 1: √𝟒𝟗𝒛𝟖 Example 2: – √(𝟐𝒂 − 𝟏)𝟔𝟑

Example 3: √81𝑛8𝑚124

Exercises:

1.√81 2.√−343𝟑

3. √144𝑝6

4.±√4𝑎10 5.√243𝑝105

6. – √𝑚6𝑛93

7.√−𝑏123

8.√16𝑎10𝑏8 9. √121𝑥6

10.√(4𝑘)4 11. ±√169𝑟4 12. – √−27𝑝63

13. – √625𝑦2𝑧4 14. √36𝑞34 15. √100𝑥2𝑦4𝑧2

16.√−0.0273

17. – √−0.36 18. √0.64𝑝10

19. √(2𝑥)84

20. √(11𝑦2)4 21. √(5𝑎2)63

22. √(3𝑥 − 1)2 23. √(𝑚 − 5)63

24. √36𝑥2 − 12𝑥 + 1

√𝑥𝑛

15

Approximate Radicals with a Calculator

Irrational Number

a number that cannot be expressed as ___________________________________________, examples:

Example: Use a calculator to approximate √𝟏𝟖. 𝟐𝟓

to three decimal places.

√18.23

≈ Use a calculator to approximate each value to three decimal places.

25. √8566

26. √0.05 27. – √5004

Solving Equations using nth Roots Round to the nearest hundredth.

31. 1

2𝑥4– 10 + 𝑥4 = 12 − 2𝑥4 32. (x – 4)3 + 4 =

1

4 33. 5 = -

3

2(x2 – 4)5

Leave your answer in simplified radical form.

34. 1

2𝑥3 +

1

2 =

3

8 35.

(2x – 9)4

2 – 8 = 4 36. 9 – (x – 3)5 =11

16

Applications Round your answers to the nearest hundredth.

1. The equation c = √𝑏25

estimates the number of collisions where b represents the number of bicycle riders per intersection.

a. Estimate the number of collisions at an intersection that has 1000 bicycle riders per week.

b. If the total number of collisions reported in one week is 21, estimate the number of bicycle riders that passed through that intersection.

2. The surface area of a sphere can be determined from the volume of the sphere using the formula

S = √36𝜋𝑉23

, where V is the volume.

a. Determine the surface area of a sphere with a volume of 200 cubic inches.

b. If the surface area of a sphere is 214.5 square inches, determine the volume.

3. Designers must create satellites that can resist damage from being struck by small particles of dust and rocks. A study showed that the diameter in millimeters d of the hole created in a solar cell

by a dust particle traveling with energy k in joules is about d = .926 √𝑘3

- 0.169. a. Estimate the diameter of a hole created by a particle traveling with energy 3.5 joules.

b. If a hole has diameter of 2.5 millimeters, estimate the energy with which the particle that made the hole was traveling.

17

6.5 Operations with Radical Expressions

Simplify the following radical expressions in two different correct ways:

1. −6√45𝑚5𝑛3 ∙ 2√5𝑚3𝑛3 2. 6√−24𝑥𝑦3 ∙ 2√𝑥𝑦4

3

Bonus:

3. What is the difference between simplifying √24𝑥𝑦 and √24𝑥𝑦3 ?

The Product Property of Radicals

For any real numbers a and b and any integer n > 1:

1. If n is even, and a and b are both ____________, then ____________________________________________.

2. If n is odd, then ____________________________________________________________.

18

Simplify the following radical expressions:

1. √8𝑥3

45𝑦5 2. √(−

81𝑥7𝑦6

4𝑧11)

3

Quotient Property of Radicals

Operations on Radicals Simplify the following radical expressions:

3. 2√50 + 4√500 − 6√125 4. (2√3 − 4√2)(√3 + 2√2)

When can you add radical expressions? How do you do so?

When can you multiply radical expressions? How do you do so?

For any real numbers a and 𝑏 ≠ 0, and any integer n > 1, if all roots are defined.

19

Warm Up: Simplify each expression.

1. 322

5 2. 2

√3+1

The Golden Patio

Ms. Abels wants to construct the most aesthetically pleasing patio possible for her house. To do so, she

wants to make sure the ratio of the sum of the length and the width to the length is the same as the ratio

of the length to the width. Ms. Abels hires the Golden Patio Company, and they build a frame with a width

of 7 meters and a length of 7+7√5

2 meters. Check to make sure that this patio meets Ms. Abels’ standards.

What is this “golden ratio” that Ms. Abels wants for her patio? Simplify as much as possible while keeping

your answer in terms of radicals.

The “golden ratio” was used to find the ratio of the length to the width of the patio. What would be the

ratio of the width to the length? Simplify your answer as much as possible while keeping it in terms of

radicals.

Just for fun! Ms. Abels’ patio is what is called a “golden rectangle.” Here are some other examples of the

Golden Ratio and golden rectangles:

20

Rationalizing Denominators

To rationalize a denominator, __________________________________________________________________________________.

Examples:

1. 3

√965 2.

√33

√53

−2

Basic Rational Exponents

Examples:

3. √16

6

√23 4.

√44

√2

5. √10

3

√6253

6.

7.

8.

𝑏𝑥𝑦 = √ = √

21

6.6 Rational Exponents

Exercises:

Write each expression in radical form, or write each radical in exponential form.

1. 111

7 2. 151

3 3. 3003

2

4. √47 5. √3𝑎5𝑏23

6. √162𝑝54

Evaluate each expression. For these, you will re-write like you did above, but then simplify.

7. −272

3 8. 2161

3 9. 451

2 10. 27−4

3

Review of Properties of Exponents

Property Definition/Example:

Product of Powers xaxb = ________

Quotient of Powers 𝑥𝑎

𝑥𝑏=__________

Negative Exponent x-a = _________

Power of a Power (xa)b = _______

Power of a Product (xy)a = _______

Power of a Quotient (x/y)a = ______

Zero Power x0 = _______

Simplify.

1. 5𝑦2

3 ∙ 3𝑦3

8 2. 𝑥3𝑦

23𝑧

−13

𝑥12𝑦

12

3. (𝑧1

2)

3

4

4. (𝑥

−12𝑦4)

14

𝑥23𝑦

32∙𝑥

−32𝑦

12

5. (𝑥3𝑦2)

32

(𝑥−1𝑦−

23)

14

22

Simplifying Radicals with Different Indices **Cannot Add/Subtract/Multiply/Divide two radicals that have a _______________ ________________**

Example: √𝑥3

√𝑥5

In order to simplify, it’s possible to re-write as a ___________ ____________, and apply the properties from yesterday.

• Since Rational Exponents and Radicals can be written in either form, give your final answer in the

______ ______ as the original problem.

• If both are present in the original problem, give answer in ____________ _____________.

1. √16

6

√23

2. √4

4

√2

3. √𝑦3 √𝑦3√𝑦

4. √𝑥 √𝑥23

√𝑥4

5. √53

∙ 252

3 ∙ √5 6.

√√ 10 ∙ √ 100

53

√ 1000 4

23

7. √ 3 3

√√ 27 ∙ √ 81 54

8.

√√553

√ √254

∙ √6256

Using Conjugates to Rationalize the Denominator

9. Simplify 𝑓(𝑥)

ℎ(𝑥) .

10.

Simplify 𝑔(𝑥)

𝑓(𝑥) .

1

2( ) 1f x y

3

4( )g x y1

2( ) 1h x y

1

2( ) 1f x x 1

2( ) 1g x x

24

6.7 Solving Radical Equations and Inequalities

1. 2√4𝑥 + 8 − 4 = 8 2. 4√2𝑥 + 113

− 2 = 10

3. √3𝑥 + 1 = √5𝑥 − 1 4. √4𝑥 = 𝑥 − 3

5. 𝑥 − 4 = √2𝑥 6. √5 − 𝑥 − 4 = 6

7. √4 + 7𝑥 = √7𝑥 − 9 8. 𝑥 = √7 − 𝑥 + 1

Important:

Important: Important:

Important:

25

Solving Equations with Rational Exponents

Warm Up: Solve 3𝑥2

3 = 27. How many solutions do you obtain?

1. 3𝑥2

3 = 27 2. 5𝑥4

3 = 80

3. 3(𝑥 + 1)4

3 = 48 4. (𝑥 − 5)2

3 = 4

5. 18 + 7𝑥1

2 = 12 6. 9 + 5(2𝑥)1

3 = 29

7. 18 − 3𝑥1

2 = 25 8. (2𝑥 + 3)1

3 = 2

Important:

Important: Important:

Important:

26

Solving Inequalities 1. 5 − √20𝑥 + 4 ≥ −3 2. 3√2𝑥 − 1 + 6 < 15

3. √𝑥 − 2 + 4 ≥ 7 4. √10𝑥 + 9 − 2 > 5

5. 8 − √3𝑥 + 4 ≥ 3 6. √2𝑥 + 8 − 4 > 2

7. 9 − √6𝑥 + 3 ≥ 6

Important: Important: