6.1 n th roots and rational exponents what you should learn: goal1 goal2 evaluate nth roots of real...
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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.
6.1 nth Roots and Rational Exponents6.1 nth Roots and Rational Exponents
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
6.1 nth Roots and Rational Exponents6.1 nth Roots and Rational Exponents
Using Rational Exponent NotationRewrite the expression using RADICAL notation.
Ex) 3 24 31
24
Ex) 4 28 41
28
6.1 nth Roots and Rational Exponents6.1 nth Roots and Rational Exponents
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
6.1 nth Roots and Rational Exponents6.1 nth Roots and Rational Exponents
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
6.1 nth Roots and Rational Exponents6.1 nth Roots and Rational Exponents
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
6.1 nth Roots and Rational Exponents6.1 nth Roots and Rational Exponents
4 625 4 625
6.26.2 Properties of Rational Exponents
What you should learn:GoalGoal 11
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
Reflection on the SectionReflection on the SectionReflection on the SectionReflection on the Section
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
What you should learn:GoalGoal 11
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.
Reflection on the SectionReflection on the SectionReflection on the SectionReflection on the Section
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
Reflection on the SectionReflection on the SectionReflection on the SectionReflection on the Section
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
Graphing Radical Functions
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
Graphing Radical Functions
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
Reflection on the SectionReflection on the SectionReflection on the SectionReflection on the Section
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
What you should learn:GoalGoal 11
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.
6.6 Solving Radical Equations6.6 Solving Radical Equations
Simple Radical
6.6 Solving Radical Equations6.6 Solving Radical Equations
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.
6.6 Solving Radical Equations6.6 Solving Radical Equations
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!!
6.6 Solving Radical Equations6.6 Solving Radical Equations
Two Radicals
02212 xx
x212 2212 x
Ex.4)
x2 x2
x2 22 xx212 x4
x612 6 6
x2
022)2(212
0228
02222
Don’t forget to Don’t forget to check your check your solutions!!solutions!!
x2 x2
6.6 Solving Radical Equations6.6 Solving Radical Equations
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)
6.6 Solving Radical Equations6.6 Solving Radical Equations
Radicals with an Extraneous Solution
xx 43 Ex.5)
6.6 Solving Radical Equations6.6 Solving Radical Equations
23x 24x
962 xx x4
Don’t forget to Don’t forget to check your check your solutions!!solutions!!
x4 x4
Radicals with an Extraneous Solution
xx 43 Ex.5)
9102 xx 0
)1)(9( xx 0
9x 1x
)9(439
)1(431
6.6 Solving Radical Equations6.6 Solving Radical Equations
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
Let’s try …Solving the Rational Exponent equation using the TI-84.
Y=
43 41 x
413x
Y= 4
Example)
Reflection on the SectionReflection on the SectionReflection on the SectionReflection on the Section
Without solving, explain why
AssignmentAssignmentPage 441Page 441
17 - 53 odd17 - 53 odd
6.6 Solving Radical Equations6.6 Solving Radical Equations
842 x has no solution.