6.1 n th Roots and Rational Exponents What you should learn: Goal1 Goal2 Evaluate nth roots of real...

6.16.1 nth Roots and Rational Exponents

What you should learn:GoalGoal 11

GoalGoal 22

Evaluate nth roots of real numbers using both radical notation and rational exponent notation

Evaluate the expression.

6.1 nth Roots and Rational Exponents6.1 nth Roots and Rational Exponents

GoalGoal 33 Solving Equations.


Using Rational Exponent NotationRewrite the expression using RATIONAL EXPONENT notation.

Ex) 3 125 31

125

Ex) 4 81 41

81

5

3

If n is odd, then a has one real nth root: naan 1

naan 1

If n is even and a = 0, then a has one nth root: 0001

nn

If n is even and a > 0, then a has two real nth roots:

If n is even and a < 0, then a has NO Real roots:

GoalGoal 11


Using Rational Exponent NotationRewrite the expression using RADICAL notation.

Ex) 3 24 31

24

Ex) 4 28 41

28


Evaluating ExpressionsEvaluate the expression.

Ex) 39 23

9

Ex)5

232

1 5

232

33 27

25 32

1

22

1

4

1

GoalGoal 22


Solving Equations

814 xEx)4 4

4 81x

3x

325 x

Ex)

5 5

5 32x

2x

642 5 x

When the exponent is EVEN you must use the Plus/Minus When the exponent

is ODD you don’t use the Plus/Minus

GoalGoal 33


Solving Equations

256)4( 4 xEx)4 4

4 2564 x

44 x

44 x 44 x

8x 0x

Very Important2 answers !

Take the Square 1st.

Reflection on the SectionReflection on the SectionReflection on the SectionReflection on the Section

Evaluate the expressions.

AssignmentAssignmentPg. 404Pg. 404

# 13 – 61 odd# 13 – 61 odd


4 625 4 625

6.26.2 Properties of Rational Exponents


GoalGoal 22

Use properties of rational exponents to evaluate and simplify expressions.

Use properties of rational exponents to solve real-life problems.

6.2 Properties of Rational Exponents6.2 Properties of Rational Exponents

Review of Properties of Exponents from section 6.1

am * an = am+n

(am)n = amn

(ab)m = ambm

a-m =

= am-n

=

n

m

a

a

ma

1

m

b

a

m

m

b

a

These all work These all work for fraction for fraction

exponents as exponents as well as integer well as integer

exponents.exponents.

Ex: Simplify. (no decimal answers)

a. 61/2 * 61/3

= 61/2 + 1/3

= 63/6 + 2/6

= 65/6

b. (271/3 * 61/4)2

= (271/3)2 * (61/4)2

= (3)2 * 62/4

= 9 * 61/2

c. (43 * 23)-1/3

= (43)-1/3 * (23)-1/3

= 4-1 * 2-1

= ¼ * ½

= 1/8

d.

= = =

3

4

1

4

1

9

18

4

3

4

3

9

18 4

3

9

18

4

3

2** All of these examples were in rational exponent form to begin with, so the answers should be in the same form!

Ex: Simplify.

a. =

= = 5

b. =

= = 2

Ex: Write the expression in simplest form.

a. = =

=

b. =

= =

=

33 525 3 5253 125

3

3

4

323

4

32

3 8

4 64 4 416 44 416

4 4 2

4

8

74

4

8

7 Can’t have a Radical in the basement!

4

4

4

4

2

2

8

7

4

4

16

14

2

144

** If the problem is ** If the problem is in radical form to in radical form to begin with, the begin with, the answer should be in answer should be in radical form as well.radical form as well.

Ex: Perform the indicated operationa. 5(43/4) – 3(43/4)

= 2(43/4)

b.

=

=

=

c.

=

=

= 33 381 33 3327

33 333 3 32

33 5625 33 55125

33 555 3 5 6

If the original problem is in radical form,

the answer should be in radical form as well.

If the problem is in rational exponent form, the answer should be in rational exponent form.

More Examples

a.

b.

c.

d.

2x x

6 6x x

11 11y y

4 8r 4 44 rr 4 44 4 rr

rr 2r

Ex: Simplify the Expression. Assume all variables are positive.

a.

b. (16g4h2)1/2

= 161/2g4/2h2/2

= 4g2h

c.

3 927z 3 93 27 z 33z

510

5

y

x5 10

5 5

y

x

2y

x

d.34

1

3

2

6

18

tr

rs33

2

4

11

3 tsr

33

2

4

3

3 tsr


Can you find the quotient of two radicals with different indices?

AssignmentAssignmentPg. Pg.

6.2 Properties of Rational Exponents6.2 Properties of Rational Exponents

Yes, Change both to rational exponent form and use the quotient property.

6.36.3 Perform Function Operations and Composition


GoalGoal 22

Perform operations with functions including power functions.

Use power functions and function operations to solve real-life problems.

6.3 Power Functions and Functions Operations6.3 Power Functions and Functions Operations

A2.2.5

Operations on FunctionsOperations on Functions: for any two : for any two functions f(x) & g(x)functions f(x) & g(x)

1.1. AdditionAddition h(x) = f(x) + g(x)

2.2. SubtractionSubtraction h(x) = f(x) – g(x)

3.3. MultiplicationMultiplication h(x) = f(x)*g(x) OR f(x)g(x)

4.4. DivisionDivision h(x) = f(x)/g(x) OR f(x) ÷ g(x)

5.5. CompositionComposition h(x) = f(g(x)) OR g(f(x))

** DomainDomain – all real x-values that “make sense” (i.e. can’t have a zero in the denominator, can’t take the even nth root of a negative number, etc.)

Ex: Let f(Ex: Let f(xx)=3)=3xx1/31/3 & g( & g(xx)=2)=2xx1/31/3. . Find (a) the sum, (b) the difference, Find (a) the sum, (b) the difference,

and (c) the domain for eachand (c) the domain for each..

(a)3x1/3 + 2x1/3 = 5x1/3

(b)3x1/3 – 2x1/3 = x1/3

(c) Domain of (a) all real numbers

Domain of (b) all real numbers

ExEx: Let f(x)=4x: Let f(x)=4x1/31/3 & g(x)=x & g(x)=x1/21/2.. Find (a) the product, (b) the quotient, and (c) the Find (a) the product, (b) the quotient, and (c) the

domain for each.domain for each.

(a) 4x1/3 * x1/2 = 4x1/3+1/2 = 4x5/6

(b)

= 4x1/3-1/2 = 4x-1/6 =

2

1

3

1

4

x

x

6

1

4

x

(c) Domain of (a) all reals ≥ 0, because you can’t take the 6th root of a negative number.

Domain of (b) all reals > 0, because you can’t take the 6th root of a negative number and you can’t have a denominator of zero.

564 x

6

4

x

CompositionComposition

• f(g(x)) means you take the function g and f(g(x)) means you take the function g and plug it in for the x-values in the function f, plug it in for the x-values in the function f, then simplify.then simplify.

• g(f(x)) means you take the function f and g(f(x)) means you take the function f and plug it in for the x-values in the function g, plug it in for the x-values in the function g, then simplify.then simplify.

Ex: Let Ex: Let ff((xx)=2)=2xx-1-1 & & gg(x)=(x)=xx22-1.-1. Find (a) Find (a) ff(g((g(xx)), (b) )), (b) gg((ff((xx)), (c) )), (c) ff((ff((xx)), and (d) the )), and (d) the

domain of each.domain of each.(a) 2(x2-1)-1 =

1

22 x

(b) (2x-1)2-1

= 22x-2-1

= 14

2

x

(c) 2(2x-1)-1

= 2(2-1x)

=2

2x x

(d) Domain of (a) all reals except Domain of (a) all reals except x=±1.x=±1.

Domain of (b) all reals except x=0.Domain of (b) all reals except x=0.

Domain of (c) all reals except x=0, Domain of (c) all reals except x=0, because 2xbecause 2x-1-1 can’t have x=0. can’t have x=0.


How is the composition of functions different form the product of functions?

AssignmentAssignmentPage Page

##

The composition of functions is a function of a function. The output of one function becomes the

input of the other function. The product of functions is the product of the output of each function when

you multiply the two functions.

6.3 Power Functions and Functions Operations6.3 Power Functions and Functions Operations

6.46.4 Inverse Functions

What you should learn:

GoalGoal 11

GoalGoal 22

Find inverses of linear functions.

Verify that f and g are inverse functions.

6.4 Inverse Functions6.4 Inverse Functions

Michigan Standard A2.2.6

GoalGoal 33 Graph the function f. Then use the graph to determine whether the inverse of f is a function.

Review from chapter 2Review from chapter 2

• Relation – a mapping of input values (x-values) onto output values (y-values).

• Here are 3 ways to show the same relation.

y = x2 x y

-2 4

-1 1

0 0

1 1

Equation

Table of values

Graph

• Inverse relation – just think: switch the x & y-values.

x = y2

xy

x y

4 -2

5 -1

0 0

1 1

** the inverse of an

equation: switch the x

& y and solve for y. ** the

inverse of a table:

switch the x & y.

** the inverse of a graph: the reflection of the original graph

in the line y = x.

To find the inverse of a function:To find the inverse of a function:

1. Change the f(x) to a y.

2. Switch the x & y values.

3. Solve the new equation for y.

** Remember functions have to pass the vertical line test!

Ex: Find an inverse of y = -3x+6.

• Steps: -switch x & y

-solve for y

y = -3x + 6

x = -3y + 6

x - 6 = -3y

yx

3

6

23

1

xy

Inverse Functions

• Given 2 functions, f(x) & g(x), if f(g(x)) = x AND g(f(x)) = x, then f(x) & g(x) are inverses of each other.

Symbols: f -1(x) means “f inverse of x”

Ex: Verify that f(x)= -3x+6 and g(x) = -1/3x + 2 are inverses.

• Meaning find f(g(x)) and g(f(x)). If they both equal x, then they are inverses.

f(g(x))= -3(-1/3x + 2) + 6

= x – 6 + 6

= x

g(f(x))= -1/3(-3x + 6) + 2

= x – 2 + 2

= x

** Because ** Because f(g(x))= xf(g(x))= x and and g(f(x)) = xg(f(x)) = x, , they are inversesthey are inverses..

Ex: (a) Find the inverse of f(x) = x5.

1. y = x5

2. x = y5

3. 5 55 yx

yx 5

5 xy

(b) Is f -1(x) a function?

(hint: look at the graph!

Does it pass the vertical line test?)

Yes , f -1(x) is a function.

Horizontal Line TestHorizontal Line Test

• Used to determine whether a function’s inverseinverse will be a function by seeing if the original function passes the horizontal horizontal line testline test.

• If the original function passespasses the horizontal line test, then its inverse is a inverse is a functionfunction.

• If the original function does not passdoes not pass the horizontal line test, then its inverse is not inverse is not a functiona function.

Ex: Graph the function f(x)=x2 and determine whether its inverse is a function.

Graph does not pass the horizontal line test, therefore the inverse is not a function.

Ex: f(x)=2x2 - 4 Determine whether f -1(x) is a function, then find the inverse equation.

2

2

4y

x

f -1(x) is not a function.

y = 2x2 - 4

x = 2y2 - 4

x + 4 = 2y2

2

4

xy

22

1 xyOR, if you fix the

tent in the basement…

Ex: g(x)=2x3

Inverse is a function!

y = 2x3

x = 2y3

3

2y

x

yx

3

2

3

2

xy

OR, if you fix the tent in the basement…

2

43 xy


Describe the steps for finding the inverse of a relation.

assignmentassignment

6.4 Inverse Functions6.4 Inverse Functions

Write the original equation; switch x and y;

solve for y

6.56.5 Graphing Square Root and Cube Root Functions.

What you should learn:

GoalGoal 11

GoalGoal 22

Graph square root and cube root functions.

Use square root and cube root functions to solve real-life problems.

6.5 Graphing Square Root and Cube Root Functions6.5 Graphing Square Root and Cube Root Functions

Michigan Standard A2.3.3

Domain: x 0, Range: y 0

Domain and range: all real numbers

Graphing Radical Functions

You have seen the graphs of y = x and y = x . These are examples of radical functions.

3

Domain: x 0, Range: y 0 Domain and range: all real numbers


You have seen the graphs of y = x and y = x . These are examples of radical functions.

3

In this lesson you will learn to graph functions of the form y = a x – h + k and y = a x – h + k.3


GRAPHS OF RADICAL FUNCTIONS

1STEP

Shift the graph h units horizontally and k units vertically.2STEP

To graph y = a x – h + k or y = a x – h + k, follow these steps.3

Sketch the graph of y = a x or y = a x .3

Comparing Two Graphs

SOLUTION

Describe how to obtain the graph of y = x + 1 – 3 from the graph of y = x .

To obtain the graph of y = x + 1 – 3, shift the graph of y = x left 1 unitand down 3 units.

Note that y = x + 1 – 3 = x – (–1) + (–3), so h = –1 and k = –3.

Graphing a Square Root Function

SOLUTION

1

2

So, shift the graph right 2 units and up 1 unit.

The result is a graph that starts at (2, 1) and passes through the point (3, –2).

Graph y = –3 x – 2 + 1.

Note that for y = –3 x – 2 + 1, h = 2 and k = 1.

Sketch the graph of y = –3 x (shown dashed). Notice that it begins at the origin and passes through the point (1, –3).

Graphing a Cube Root Function

SOLUTION

1

2

So, shift the graph left 2 units and down 1 unit.

The result is a graph that passes through the points (–3, –4), (–2, –1), and (–1, 2).

Graph y = 3 x + 2 – 1.3

Sketch the graph of y = 3 x (shown dashed). Notice that it passes through the origin and the points (–1, –3) and (1, 3).

3

Note that for y = 3 x + 2 – 1, h = –2 and k = –1.

3

Finding Domain and Range

State the domain and range of the functions in the previous examples.

SOLUTION

From the graph of y = –3 x – 2 + 1, you can see that the domain of the function is x 2 and the range of the function is y 1.

Finding Domain and Range

State the domain and range of the functions in the previous examples.

SOLUTION

SOLUTION

From the graph of y = –3 x – 2 + 1, you can see that the domain of the function is x 2 and the range of the function is y 1.

From the graph of y = 3 x + 2 – 1, you can see that the domain and range of the function are both all real numbers.

3

Using Radical Functions in Real Life

When you use radical functions in real life, the domain is understood to be restricted to the values that make sense in the real-life situation.

The model that gives the speed s (in meters per second) necessary to keep a person pinned to the wall is

where r is the radius (in meters) of the rotor. Use a graphing calculator to graph the model. Then use the graph to estimate the radius of a rotor that spins at a speed of 8 meters per second.

s = 4.95 r

AMUSEMENT PARKS At an amusement park a ride called the rotor is a cylindrical room that spins around. The riders stand against the circular wall. When the rotor reaches the necessary speed, the floor drops out and the centrifugal force keeps the riders pinned to the wall.

AMUSEMENT PARKS At an amusement park a ride called the rotor is a cylindrical room that spins around. The riders stand against the circular wall. When the rotor reaches the necessary speed, the floor drops out and the centrifugal force keeps the riders pinned to the wall.

Modeling with a Square Root Function

SOLUTION

You get x 2.61.

The radius is about 2.61 meters.

Graph y = 4.95 x and y = 8. Choose a viewing window that shows the point where the graphs intersect. Then use the Intersect feature to find the x-coordinate of that point.

Modeling with a Cube Root Function

Use a graphing calculator to graph the model. Then use the graph to estimate the age of an elephant whose shoulder height is 200 centimeters.

SOLUTION

The elephant is about 8 years old.

You get x 7.85

Biologists have discovered that the shoulder height h (in centimeters) of a male African elephant can be modeled by

h = 62.5 t + 75.83

where t is the age (in years) of the elephant.

Graph y = 62.5 x + 75.8 and y = 200 with your calculator. Choose a viewing window that shows the point where the graphs intersect. Then use the Intersect feature to find the x-coordinate of that point.

3


Give an example of a radical function.

assignmentassignment

7.5 Graphing Square Root and Cube Root Functions7.5 Graphing Square Root and Cube Root Functions

233 xy

6.66.6 Solving Radical Equations


GoalGoal 22

Solve equations that contain

Radicals.

6.6 Solving Radical Equations6.6 Solving Radical Equations

Solve equations that contain

Rational exponents.

Michigan Standard L1.2.1

Solve the equation. Check for extraneous solutions.

3x

223x

9x

check your check your solutions!!solutions!!

GoalGoal 11 Solve equations that contain Radicals

Ex.1)Key Step:

To raise each side of the equation to the same power.


Simple Radical


1263 x

3 x

33 x

Ex.2)

6 6

6

36x 216

Key Step:

Before raising each side to the same power, you should isolateisolate the radical expression on one side of the

equation.

Simple Radical

Don’t forget

to check

it.


One Radical

6482 x

82 x 282 x

Ex.3)

4 4

10 210

82 x 1008 8

922 x2 2

46x

648)46(2

64100

6410

Don’t forget to Don’t forget to check your check your solutions!!solutions!!


Two Radicals

02212 xx

x212 2212 x

Ex.4)

x2 x2

x2 22 xx212 x4

x612 6 6

x2

022)2(212

0228

02222


x2 x2


Radicals with an Extraneous Solution

What is an Extraneous Solution?

… is a solution to an equation raised to a power that is not a solution to the original equation.

xx 43 Example 5)



xx 43 Ex.5)


23x 24x

962 xx x4


x4 x4


xx 43 Ex.5)

9102 xx 0

)1)(9( xx 0

9x 1x

)9(439

)1(431


GoalGoal 22 Solve equations that contain Rational exponents.

2 2

250223

xEx. 6)

23x 125 3223x 32125

x 231125

x 25

x 25it

250)25(223

21)15625(2

213)25(2

)125(2 250

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Let’s try …Solving the Rational Exponent equation using the TI-84.

Y=

43 41 x

413x

Y= 4

Example)


Without solving, explain why

AssignmentAssignmentPage 441Page 441

17 - 53 odd17 - 53 odd


842 x has no solution.

6.1 n th Roots and Rational Exponents What you should learn: Goal1 Goal2 Evaluate nth roots of real...

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