Quiz 6-4Quiz 6-41.1.

2.2.

4. Are the following functions inverses of 4. Are the following functions inverses of each other ? (hint: you must use a each other ? (hint: you must use a compositioncomposition to prove it).to prove it).

(-2, 5), (5, 6), (-2, 6), (7, 6)(-2, 5), (5, 6), (-2, 6), (7, 6)

What is the inverse relation of:What is the inverse relation of:

Find the inverse of: y = 0.5x + 2Find the inverse of: y = 0.5x + 2

?)(1 xf2)( xxf

3.3.

42)( xxf 22)( xxg

3 2x

y

Quiz 6-2 Quiz 6-2 Rationalize the DenominatorRationalize the Denominator

1. 1.

3. 3.

62

32. 2.

4 32zy

x

The Square Roof FunctionThe Square Roof Function(Input)(Input) xx

(rule)(rule)

(output)(output) f(x)f(x)

00 00

11

xxf )(

1

22 41.1

73.133

244

24.255

45.266

65.277

+ + yy

+ + xx

2

1 5

3

4

6

1

2

0

7

888 83.2

3

Section 6-5Section 6-5 Graphing Radical Equations.Graphing Radical Equations.

322 xy

xy Your turn:Your turn:Graph the following on your calculator:Graph the following on your calculator:

1. 1. What is the What is the domaindomain of the function? of the function?

2. 2. What is the What is the rangerange of the function? of the function?

3. 3. (don’t delete the first equation): graph the following then(don’t delete the first equation): graph the following then select the correct answer from the 3 following choices:select the correct answer from the 3 following choices:

a. It is the graph of shifted up 2, and left 3. a. It is the graph of shifted up 2, and left 3. xy

b. It is the graph of shifted down 3, and right 2. b. It is the graph of shifted down 3, and right 2. xy

xy c. It is the graph of shifted down 3, and left 2. c. It is the graph of shifted down 3, and left 2.

Section 6-6Section 6-6

Solve Radical FunctionsSolve Radical Functions

VocabularyVocabulary

Radical EquationRadical Equation: An equation with a radical : An equation with a radical symbol in it. symbol in it. 3 25 x

Review (Solving single Review (Solving single variable equations)variable equations)

10 = 3x – 2 10 = 3x – 2

What does it mean to solve an equation ?What does it mean to solve an equation ?

““use properties of equality to get theuse properties of equality to get the variable on one side of the equal signvariable on one side of the equal sign and every other number on the other side.”and every other number on the other side.”

+ 2+ 2+ 2+ 2

12 = 3x12 = 3x

÷ 3÷ 3÷ 3÷ 34 = x4 = x

Solving an Exponential Solving an Exponential EquationEquation(Review Section 6-2)(Review Section 6-2)

8)4(2 2 x

4)4( 2 x÷2÷2 ÷2÷2

21

21

2 4)4( x

Isolate the base and itsIsolate the base and its exponent.exponent.

““undo” squaringundo” squaring

(turn the exponent (turn the exponent into a ‘1’)into a ‘1’)

24 x Even Root Even Root !!!!!! ± ± resultresult

24x x = 6, 2x = 6, 2

Your Turn:Your Turn:

Solve for ‘x’Solve for ‘x’

4.4. 25)3( 2 x

5. 5. 27)1( 3 x

Now solve a radical Now solve a radical equation.equation.

Remove the radical Remove the radical by squaring each sideby squaring each side

36 x

2236 x

96x-6-6 -6-6

x = 3x = 3

Isolate ‘x’ on one side Isolate ‘x’ on one side of the equal sign by of the equal sign by subtracting 6 from subtracting 6 from bothboth sides. sides.

Using Exponent Form (may be Using Exponent Form (may be easier)easier)

36 x ““undo” square root undo” square root square both sides square both sides

Solve for ‘x’Solve for ‘x’96xx = 3x = 3

2236 x

Your Turn:Your Turn:

6. 6.

7.7.

43 x

253 x

Another example:Another example:

1153 x

253 x

-1-1 -1-1Subtract ‘1’ from each sideSubtract ‘1’ from each side

Cube each sideCube each side

333 25 x

85 x3x

Add ‘5’ to each sideAdd ‘5’ to each side

Example:Example:

1233 x

Same thing as:Same thing as:

1233 x

1233 x Add ‘2’ to both sidesAdd ‘2’ to both sides

333 x ““cube” both sidescube” both sides

273x Subtract “3” from both sidesSubtract “3” from both sides

24x

Your Turn:Your Turn:

8. 8.

9.9. 8212 x

5253 x

This will be written as:This will be written as:

5253 x

10.10. 102103 x

Solve rational equations of Solve rational equations of Exponent formExponent form

162 23

x Isolate the ‘x’ term Isolate the ‘x’ term divide by ‘2’ divide by ‘2’

2/3 root 2/3 root but use exponents to do it but use exponents to do it

How do you “undo” 3/2 power?How do you “undo” 3/2 power?823

x

323

2

23

8

x

328x Convert to radical form to simplifyConvert to radical form to simplify

23 8x 23 2*2*2 422

Solve rational equations of Solve rational equations of Exponent formExponent form

729 53

x Isolate the ‘x’ term Isolate the ‘x’ term divide by ‘9’ divide by ‘9’

5/3 root 5/3 root but use exponents to do it but use exponents to do it

How do you “undo” 3/5 power?How do you “undo” 3/5 power?853

x

353

5

53

8

x

358x Convert to radical form to simplifyConvert to radical form to simplify

53 8x 53 2*2*2 3225

Your Turn:Your Turn:

11. 11.

12.12.

162

1 25

x

542 32

x

More complicated:More complicated:53212 3

4 x Add ‘21’ to both sidesAdd ‘21’ to both sides

Divide both sides by “-2” Divide both sides by “-2” 322 34

x

1634

xNeed an exponent of ‘1’ on ‘x’Need an exponent of ‘1’ on ‘x’ raise both sides to the ¾ power.raise both sides to the ¾ power.

434

3

34

16

x

43

16x44thth root of 16 raised to the 3 root of 16 raised to the 3rdrd power power

34 )16(x 34 )2*2*2*2( 823

Your Turn:Your Turn:

13. 13. 1643 32

x

14.14. 212 41

x

15.15. 1502253 37

x

VocabularyVocabulary

Extraneous SolutionExtraneous Solution: a solution that, when : a solution that, when plugged back into the original equation,plugged back into the original equation, does not make a true statement.does not make a true statement.

xxx 3106410

xx 464

xx 6

Get the radical all by itselfGet the radical all by itself

Square both sidesSquare both sides 22)6( xx

26 xx

Subtract ‘x’ and ’10’ from Subtract ‘x’ and ’10’ from both sides.both sides.

Divide both sides by ‘4’.Divide both sides by ‘4’.

Non standard formNon standard form quadratic quadratic

Equations with Equations with twotwo solutions solutions

FactorFactor062 xx

Solve using Solve using zero product propertyzero product property

26 xx Write quadratic in standard form.Write quadratic in standard form.

0)2)(3( xx

2,3 x Check for Check for extranious solutions. extranious solutions.

Check: x = 3Check: x = 3 in the original equation. in the original equation.

xxx 3106410

2,3 x

)3(36343

)2(36242

Check for extraneous solutions.Check for extraneous solutions.

Check: x = 3Check: x = 3

??9)3(43 ??

x = 3 is x = 3 is extraneousextraneous

Check: x = -2Check: x = -2??

)2(3442 ??

x = -2 is the x = -2 is the onlyonly solution solution

xxx 3106410 xxx 364

6)2(42 ??

Your turn:Your turn:

16. 16. xx 910

17. 17. 22

32 xx

Equations with Equations with twotwo radicals radicals(the easy version)(the easy version)

xx 253 Square left/right sidesSquare left/right sides

xx 253

533 x

Add 2x left/rightAdd 2x left/right

Subtract 3 left/rightSubtract 3 left/right

23 x divide 3 left/rightdivide 3 left/right

32x

Your turn: Your turn: SolveSolve

18. 18. 5542 xx

19. 19. xx 7432

How would you multiply this How would you multiply this out?out?

5151 xx

251x

)1( x

FOILFOIL

15 x 15 x 25

26110 xx

3232 xx

92323)2( xxx

Your turn:Your turn: 20. 20. Simplify the product.Simplify the product.

726 xx

Equations with Equations with twotwo radicals radicals

xx 31026 Since they are not “like” radicalsSince they are not “like” radicals (same radicand and index number) (same radicand and index number) you can’t just combine them.you can’t just combine them.

If we got rid of If we got rid of oneone of the radicals (by squaring of the radicals (by squaring it) then we could then solve the equation.it) then we could then solve the equation.

2231026 xx Square both sides.Square both sides.

xxx 3102626

Equations with Equations with twotwo radicals radicals

xxx 3102626

xxxx 310462626

xxx 3106410

F.O.I.L.F.O.I.L.

Combine like termsCombine like terms

Equations with Equations with twotwo radicals radicalsxxx 3106410 Subtract 10 (left/right)Subtract 10 (left/right)

xxx 364 Subtract x (left/right)Subtract x (left/right)

xx 464 divide by 4 (left/right)divide by 4 (left/right)

xx 6 Square (left/right)Square (left/right)

26 xx Subtract ‘x’ and ‘6’ (left/right)Subtract ‘x’ and ‘6’ (left/right)

60 2 xx Factor and solve:Factor and solve:

)2)(3(0 xx x = 3, -2x = 3, -2 Check the solutionCheck the solution

2152 xx

Your turn:Your turn: 21. 21. solvesolve

HOMEWORKHOMEWORK

Section 6-6Section 6-6 BlogBlog