algebra-2 weak area review part 2. your turn: 1. which of the 3 functions restrict domain? 2. which...
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Algebra-2Algebra-2Weak Area Review Weak Area Review
Part 2Part 2
Your turn: Your turn: 1. 1. Which of the 3 functions restrict domain?Which of the 3 functions restrict domain?
xxf
1)( 2)( xxf
2. 2. Which of the 3 functions restrict range?Which of the 3 functions restrict range?
What What can’tcan’t the the radicandradicand equal ?equal ?(for even index numbers)(for even index numbers)
2)( xxf
The The eveneven root of a negative number does root of a negative number does not have a real solution.not have a real solution.
02 x 2x
The The radicandradicand (of an even root) cannot be a (of an even root) cannot be a negative number in order to have a real solution.negative number in order to have a real solution.
Your turn:Your turn:
6232)( xxf
062 x62 x
3. 3. What is the domain?What is the domain?
3x4. 4. What is the range?What is the range? 2y
Function NotationFunction Notation
y = f(x) “y is a function of x”y = f(x) “y is a function of x”
‘‘y’ equals ‘f’ of ‘x’y’ equals ‘f’ of ‘x’
A function is a A function is a rulerule that matches input that matches inputvalues to out put values.values to out put values.
f(x) = 2x + 1f(x) = 2x + 1(Input)(Input) xx
(rule)(rule) 2x + 12x + 1
(output)(output) yy
22 22(2) (2) + 1+ 1 55
f(2) = 5f(2) = 5
33 22(3) (3) + 1+ 1 77 f(3) = 7f(3) = 7
FunctionsFunctionsf(x) = 2xf(x) = 2x f(3) = ?f(3) = ?
Means: wherever you see an ‘x’ in the Means: wherever you see an ‘x’ in the function, replace it with a 3.function, replace it with a 3.
f(3) = 2(3)f(3) = 2(3)
1. Replace the ‘x’ with a set of parentheses.1. Replace the ‘x’ with a set of parentheses.
f(3) = 2( )f(3) = 2( )
2. Put the input value ‘3’ into the parentheses.2. Put the input value ‘3’ into the parentheses.
3. Find the output value.3. Find the output value.
f(3) = 6f(3) = 6
FunctionsFunctions f(2) = ?f(2) = ?
Means: wherever you see an ‘x’ in the Means: wherever you see an ‘x’ in the function, replace it with a ‘2’.function, replace it with a ‘2’.
1. Replace the ‘x’ with a set of parentheses.1. Replace the ‘x’ with a set of parentheses.
2. Put the input value ‘-2’ into the parentheses.2. Put the input value ‘-2’ into the parentheses.
3. Find the output value.3. Find the output value.
f(2) = 0f(2) = 0
23)( 2 xxxf
23)( 2 xf
2232)( 2 xf
Cool, we found a Cool, we found a zero of the function.zero of the function.
Your turn:Your turn:
5. 5. 1)( 3 xxf
6. 6. 21
2)( xxf
7. 7. 20
)4(2)(
2
xx
xxf
?)2( f
?)4( f
?)2( f
Composition of FunctionsComposition of Functions
If your input is a If your input is a functionfunction instead of a number input the expression into theinstead of a number input the expression into the function and just apply the rule to it.function and just apply the rule to it.
f(x) = 2x + 1f(x) = 2x + 1
(Input)(Input)
(rule): f(x)(rule): f(x) 2x + 12x + 1
(output)(output)
f(1) = 3f(1) = 3g(x)g(x) 22((g(x)g(x)) ) + 1+ 1
1 – 3x 1 – 3x f(f(g(x)g(x))= )= f(f(1 – 3x1 – 3x) = 3 – 6x ) = 3 – 6x
g(x) = 1 – 3xg(x) = 1 – 3x
11 22(1) (1) + 1+ 1 33
22((11 – 3x– 3x) ) + 1+ 1 3 – 6x 3 – 6x
CompositionsCompositions of Functions of Functions2)( xxg
)( 2xf 3)(2 2 x
f(x) = 2x + 3 andf(g(x)) = ?f(g(x)) = ?
32))(( 2 xxgf
3(..)2(..) f 1.1. Replace the ‘x’ in f(x) withReplace the ‘x’ in f(x) with a set of parentheses.a set of parentheses.
2.2. Put the input value (g(x))Put the input value (g(x)) into the parentheses.into the parentheses.
1.1. The input value to f(x) is g(x).The input value to f(x) is g(x).
3. Find the output value.3. Find the output value.
CompositionsCompositions of Functions of Functions2)( xxg f(x) = 2x - 3 and
2)32())(( xxfg
2(..)(..) g
)32)(32())(( xxxfg
1.1. Replace the ‘x’ in g(x) withReplace the ‘x’ in g(x) with a set of parentheses.a set of parentheses.
3. Put the input value (f(x))3. Put the input value (f(x)) into the parentheses.into the parentheses.
2. The input value is f(x).2. The input value is f(x).
4. Find the output value.4. Find the output value.
Another way to write a composition.Another way to write a composition. ))(( xgfgf
?fg
Your turn:Your turn:
8. 8.
9. 9.
10. 10.
14)( xxf
35)( xxg
?gf
?fg
?))(( xgg
Function compositionsFunction compositions2)( xxg xxf 3)(
))2((gfgf One more layer.One more layer.
4)2()2( 2 g
12))2(( gf
The input to g(x) is 2.The input to g(x) is 2.
)4(3)4( fThe input to f(x) is g(2), so theThe input to f(x) is g(2), so the input to f(x) is 4.input to f(x) is 4.
Your turn:Your turn:
12. 12.
25)( xxf
1)( 3 xxg
?)1( gf
11. 11. ?)2( fg
13. 13. ?))3(( gf
VocabularyVocabulary
Inverse RelationInverse Relation: A relation that interchanges: A relation that interchanges the input and output values of the original relation.the input and output values of the original relation.
(-2, 5), (5, 6), (-2, 6), (7, 6)(-2, 5), (5, 6), (-2, 6), (7, 6)RelationRelation::
Inverse RelationInverse Relation:: (5, -2), (6, 5), (6, -2), (6, 7)(5, -2), (6, 5), (6, -2), (6, 7)
Graphs of Inverse Graphs of Inverse RelationsRelations y = xy = x
Each point in the inverse relation Each point in the inverse relation is a point from the relation is a point from the relation reflected across the line y = x reflected across the line y = x
How to find the inverse How to find the inverse relationrelation
Relation: Relation:
1. Exchange ‘x’ and ‘y’ in the original relation.1. Exchange ‘x’ and ‘y’ in the original relation.
2. Solve for ‘y’ (get ‘y’ all by itself).2. Solve for ‘y’ (get ‘y’ all by itself).
2 xy
2 yx
22 yx
22 xy
22 2 yx
Your Turn:Your Turn: Find the inverse of:Find the inverse of:
This is the inverse of: y = 4x + 2This is the inverse of: y = 4x + 2
14. 14. y = 4x + 2 y = 4x + 2
24 xx
yx 42
Exchange ‘x’ and ‘y’Exchange ‘x’ and ‘y’
subtract ‘2’ (left and right)subtract ‘2’ (left and right)
Divide (all of the) left and right by 4Divide (all of the) left and right by 4
4
4
4
2
4
yx
yx
2
1
4
Reduce the fractionsReduce the fractions
Rearrange into “slope intercept form”Rearrange into “slope intercept form”
2
1
4x
y
Your Turn:Your Turn: Find the inverse of:Find the inverse of:
32 xy15.15.
32 yx23 yx
Exchange ‘x’ and ‘y’Exchange ‘x’ and ‘y’
Add ‘3’ (left and right)Add ‘3’ (left and right)
Square root both sidesSquare root both sides
23 yx
3 xy This is the inverse of: This is the inverse of:
SimplifySimplify
yx 3 Which is the same thing as:Which is the same thing as:
32 xy
How to write: “the inverse of How to write: “the inverse of f(x)”f(x)” 2)(1 xxf2)( xxf
means “the inverse function of f(x)means “the inverse function of f(x)
Do not confuse this Do not confuse this notationnotation with the negative with the negative inverse property: inverse property:
11 1
xx
)(1 xf
The inverse of a The inverse of a numbernumber means “flip the means “flip the number (the reciprocal of the number)”number (the reciprocal of the number)”
The inverse of a The inverse of a functionfunction means means “exchange ‘x’ and ‘y’ then solve for ‘y’.” “exchange ‘x’ and ‘y’ then solve for ‘y’.”
Your Turn:Your Turn:
16. 16. Are f(x) and g(x) inverses of each other ?Are f(x) and g(x) inverses of each other ?
4
1)(
x
xg 14)( xxf
17. 17. Are f(x) and g(x) inverses of each other ?Are f(x) and g(x) inverses of each other ?
5
1)(
2x
xg xxf 51)(
Your Turn:Your Turn:
18. 18. SolveSolve 416 x
19. 19. SolveSolve 322 x
313 4 x
4 44 16 x 2x
22322 x
922 x
72 x
27x
Your Turn:Your Turn:20.20. Find the inverse Find the inverse
)1(log2)( 3 xxf
)1(log2 3 xy
)1(log2 3 yx
)1(log2 3 yx
)1(log2 333 yx
13 2 yx
13 2 x
y
Log = Log = = =
)1(log2 3 yx 13 2 yx
Your Turn:Your Turn:
223)( xxf
21. 21. ?)( 1 xf
223 xy
223 yx
223 yx
22223 yx
2232
2
yx
yx
229
2
yx
2*2
12
92
1 2
yx
118
2
Your Turn:Your Turn:
21. 21.
82)( )5( xexf
?)( 1 xf
82 )5( xey
82 )5( yex
)5(28 yex
)5(
2
8 yex
)5(ln2
8ln
yex
52
8ln
yx
yx
2
8ln5
2
8ln5
xy
How can you tell if the How can you tell if the inverse of a function is a inverse of a function is a function?function?
2)( xxf Horizontal Line TestHorizontal Line Test: if the line : if the line intersects the graph more than intersects the graph more than once, then the once, then the InverseInverse of the function is of the function is NOTNOT a a function.function.
Your turn: Your turn: Calculate ‘x’.Calculate ‘x’.
23.23. x7log8
Your turn:Your turn:ExpandExpand
CondenseCondense xyx 55 log7log3
33
4
logy
x24.24.
25.25.
Your turn:Your turn:
26. 26. The front row of a rock concert has a sound intensity The front row of a rock concert has a sound intensity
ofof
The reference sound intensityThe reference sound intensity
What is the sound level in decibels on the front row of the rock concert? What is the sound level in decibels on the front row of the rock concert?
21 er watts/met101
0
log10decibels)(in )(I
IIL
2120 er watts/met101 I
12
1
10
10log10)(
IL)12(110log10
1110log10 10log*11*10 db110
Your turn:Your turn:A solution has hydrogen-ion concentration of A solution has hydrogen-ion concentration of
moles/liter. What is the pH of the moles/liter. What is the pH of the
solution?solution?
pH = -log [H+]pH = -log [H+]
]100.7log[ 14pH 2.13pH
14100.7 27. 27.
Your turn:Your turn: 65log93 ds
FormulaFormula relating distance (d) that a tornado travels relating distance (d) that a tornado travels and the wind speed (s) inside the cone of the tornado.and the wind speed (s) inside the cone of the tornado.
28. 28. Some storm chasers measured the speed of the wind Some storm chasers measured the speed of the wind inside a tornado. It was 275 mph. How far will the tornado inside a tornado. It was 275 mph. How far will the tornado travel along the ground?.travel along the ground?.
(1) Plug numbers into the formula(1) Plug numbers into the formula 65log93275 d(2) Solve for the unknown variable in the formula(2) Solve for the unknown variable in the formula
100log93210
dlog93210 d93
21010 milesd 181
Your turn:Your turn: trPtA )1()( A bank account earning 4% interest has $3569 in it. The A bank account earning 4% interest has $3569 in it. The original deposit was $2000. How long has the money original deposit was $2000. How long has the money been in the account? been in the account?
29. 29.
t)04.01(20003569 t04.17845.1
Log = Log = = =
t7845.1log 04.1
t
)04.1ln(
7845.1ln
7.14t
Using an inverse function to solve an Using an inverse function to solve an equation.equation.
192.035tP
Ticket prices in the NFL can be modeled by:Ticket prices in the NFL can be modeled by:
where ‘t’ is the number of where ‘t’ is the number of years since 1995.years since 1995.
During what year was the price of a ticket $50.85 ?During what year was the price of a ticket $50.85 ?
(price as a function of time since 1995)(price as a function of time since 1995)
192.035tP
192.0
35
85.50t t
192.0
1
35
85.50
192.03585.50 t
6t 200161995
Your Turn:Your Turn:30. 30. Solve:Solve:
4)1(log2 3 x
31. 31. solve:solve: 7211 )5( xe
2)1(log3 x2)1(log 33 3 x 91x 8x
)5(218 xe
)5(9 xe
)5(ln9ln xe
5197.2 x
803.2x
Your Turn:Your Turn:
77AA
CC
BB 32. 32.
33. 33.
33
?Cm
5.75.7
FF
DD
EE
?Fm
3.23.2
Your turn: Your turn: describe the transformations of f(x)describe the transformations of f(x)
2)3sin(5.0)( xxf
33. 33. Period = ?Period = ?
34. 34. Horizontal translation (phase shift) = ?Horizontal translation (phase shift) = ?
35. 35. Vertical translation = ?Vertical translation = ?
36. 36. Amplitude = ? Amplitude = ?
37 37 frequency = ? frequency = ?