quiz 6-4 1. 2. 4. are the following functions inverses of each other ? (hint: you must use a...
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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
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