Chapter 7Radical Equations

Lesson 7.1Operations of Functions

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Operations with functions• (f + g)(x)= f(x) + g(x)

• (f – g )(x) = f(x) – g(x)

• (f · g)(x) = f(x) · g(x)

• (f/g)(x) = f(x) , g(x) = 0 g(x)

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Examples

1. f(x)= x2 – 3x + 1 and g(x) = 4x + 5 find a. (f + g)(x) b. (f – g)(x)

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2. f(x) = x2 + 5x – 1 and g(x) = 3x – 2 finda. (f · g)(x) b. (f/g)(x)

Composition of Functions

•[f o g](x) = f[g(x)]

▫Plug in the full function of g for x in function f

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Examples

3. Find [f o g](x) and [g o f](x) for f(x) = x+3 and g(x) = x2 + x – 1

4. Evaluate [f o g](x) and [g o f](x) for x = 2

TOD:Find the sum, difference, product and quotient of f(x)

and g(x) 1. f(x) = x2 + 3 g(x) = x – 4

Find [g o h](x) and [h o g](x)2. g(x) = 2x 3. h(x) = 3x – 4

If f(x) = 3x, g(x) = x + 7 and h(x) = x2 find the following

4. f[g(3)] 5. g[h(-2)] 6. h[h(1)]

Lesson 7.2 Inverse Functions and Relations

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Inverse Relations

•Two relations are inverse relations if and only if one relation contains the element (a, b) the other relation contains the element (b, a)

▫Example: Q and S are inverse relationsQ = {(1, 2), (3, 4), (5, 6)}S = {(2, 1), (4, 3), (6, 5)}

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Examples

1. Find the inverse relation of {(2,1), (5,1), (2,-4)}

2. Find the inverse relation of {(-8,-3), (-8,-6), (-3,-6)}

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Property of Inverse Functions

•Suppose f and f-1 are inverse functions. Then f(a) = b if and only if f -1 (b) = a

To find the inverse of a function: 1. Replace f(x) with y2. Switch x and y3. Solve for y4. Replace y with f-1(x)5. Graph f(x) and f-1(x) on the same

coordinate plane

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To Graph a function and it’s inverse

1. Make an x/y chart for f(x) then graph the points and connect the dots

2. Make an x/y chart for f-1(x) by switching the x and y coordinates and then graph and connect the dots

- The graphs should be reflections of one another over the line y=x

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Examples

3. Find the inverse of and then graph the function and its inverse

5

3)(

x

xf

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4. Find the inverse of and then graph the function and its inverse f(x) = 2x - 3

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Inverse Functions

•Two functions f and g are inverse functions if and only if both of their compositions are the identity function. [f o g](x) = x and [g o f](x) = x

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5. Determine if the functions are inverses

25

1)(

105)(

xxg

xxf

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6. Determine if the functions are inverses

43

1)(

33)(

xxg

xxf

Lesson 7.3Square Root Functions and Inequalities

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Square Root Functions

• Definition: when a function contains a square root of a variable

• The inverse of a quadratic function (starts with x2) is a square root function only if the range has no negative numbers!!

xy

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Parent Functionsxy 2xy

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Graphing Square Root Functions

1. Find the domain. The radicand (the stuff inside the square root) cannot be negative so take whatever is inside and make it greater than or equal to 0 and solve.

2. Plug the x value you found back in and solve for y.

3. Make a table starting with the ordered pair you found in steps 1 & 2. Graph.

4. State the range.

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Examples1. Graph. State the domain, range, and x- and y-

intercepts.

43 xy

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2. Graph. State the domain, range, and x- and y- intercepts.

532 xy

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3. Graph. State the domain, range, and x- and y- intercepts

12

3 xy

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Square Root Inequalities

•Follow same steps as an equation to graph but add last step of shading.

•Remember rules for solid and dotted lines!!

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Examples4. Graph 62 xy

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5. Graph 1 xy

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6. Graph

153 xy

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7. Graph

422 xy

7.4Nth Roots

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Nth Roots

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Symbols and Vocabulary

n #index

Radical Sign

Radicand

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More Vocabulary

• Principal Root: the nonnegative root▫ Example: 36 has two square roots, 6 and -6

6 is the principal root because it is positive

Other things to remember:- If the radical sign has a – in front of it this indicates the opposite of the principal square root- If the radical has a ± in front of it then you give both the principal and the opposite principal roots

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Summary of Nth Roots

n b > 0 b < 0 b = 0

evenone positive root, one negative root

no real roots

One real root = 0

oddone positive root, no negative root

no positive roots, one

negative root

n b n b

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Examples

425.1 x

5 201532.3 yx

82 )2(.2 y

9.4

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Your turn…

681.5 y

6 1830729.7 yx

12)3(.6 x

25.8

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More things to remember…

• When you find the nth root of an even power and the result is an odd power, you must take the absolute value of the result to ensure that the answer is nonnegative▫

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Examples Using Absolute Values

8 8.9 x 4 12)1(81.10 a

10100.11 x14)1(64.12 y

7.5Operations with Radical Expressions

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Review of Properties of Radicals

Product Property

If all parts of the radicand are

positive- separate each part so that it

has the nth root

Ex:

Quotient Property

If all parts of the radicand are

positive- separate each part so that it

has the nth root

Ex:

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When is a radical simplified?

-The nth root is as small as possible

-The radicand has no fractions

-There are no radicals in the denominator

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Rationalizing the Denominator

• If you have a fraction with a radical in the denominator you must rationalize the denominator, multiply the numerator and denominator by the square root in the denominator.

33

2

27

4

27

4

9

32

33

32

3

3

33

2

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Examples

7816.1 qp

10536.3 sr

5

4

.2y

x

7

9

.4n

m

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Simplify all radicals then combine -Remember you can only combine like terms!

1086272125.7

18432583.6

482273122.5

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Foil- then simplify

356224.10

635635.9

323253.8

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Multiply by the conjugate- then simplify

56

523.13

25

24.12

35

31.11

7.6Rational Exponents

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Fractions, Fractions, are our Friends!

• What do we do when we have a fraction as an exponent?▫ Change it into a radical

The denominator of becomes the index of the radicalThe numerator becomes the power for the radicand

Ex: = 3

1

83 8

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Examples

5

1

4

1

.2

.1

x

a

5

3

243.4

16.3 4

1

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Generalization for numerator greater than 1

3 28 23 8 22

The denominator becomes the index for the radical (nth root)

If the numerator is bigger than 1:

The numerator becomes the power for the nth root of the radicand

Ex: = = = 3

2

8

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Examples

4

3

5

7

5

1

.2

.1

y

xx

5

4

4

9

4

1

.4

.3

r

aa

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1

1.7

9.6

3

81.5

2

1

2

1

4 2

6

8

m

m

z

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2

2.10

16.9

2

32.8

2

1

2

1

3 4

3

4

y

y

x

7.7Solving Radical Equations and Inequalities

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Vocabulary

•Radical equations/inequalities: equations/inequalities that have variables in the radicands

•Extraneous Solution: when you get a solution that does not satisfy the original equation.

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Steps to solve radical equations/inequalities

1. Isolate the radical.▫ If there is more than one, isolate the radical that has the

most stuff in it first!

2. Raise both sides to the power that will eliminate the radical.▫ If there is more than one radical you will have to repeat

this step until there are no more radicals in the problem

3. Solve for the variable.4. Test solutions to check for extraneous

roots.

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Solve Radical Equations

421 x

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xx 315

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02)15(3 3

1

n

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0262 4

1

y

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Day #2Now let’s look at inequalities

•Steps to Solve a radical inequality1. Identify the values of x for which the root is

defined.2. Solve given inequality by isolating and then

eliminating the radical3.Test values to confirm your solution (use a

table to do this)

4. Graph solution on a number line. (Remember open and closed dots)

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6442 x

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5122 x

4244 x

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