Download - Inverse Functions 12.1 SAT Question: Let and for all integers x and y. If, what is the value of ?
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Inverse Functions
12.1
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SAT Question:
Let and for all integers x
and y. If , what is the value of ?
2 1
2
xx
3
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2m m
13.
85
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. 5
37.
2
A
B
C
D
E
3(2)2 3
2m
23 1 103 5
2 2
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The inverse of putting on yoursocks and then your shoes istaking off your shoes and then your socks. The beginning and end of each is the same: bare feet!
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Start with x (bare feet). Add 5 and then subtract 5.
What do you have?
x. Therefore adding and subtracting are inverses.
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What is the inverse of multiplication?
Division. What is the inverse of squaring?
Square rooting. What is the inverse of cube root?
Cubing.
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The inverse of a set of ordered pairs is found by exchanging x and y.
Given A = (2,4), (6,8), (10,12)
The inverse of A = _____________________
Is this a function?
(4,2), (8,6), (12,10)
Yes
Every function has an inverse, but the inverse is not necessarily a function. If the inverse is also a function, it is denoted by and is read “f inverse.”
1( )f x
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The inverse of (1, 4) is (4,1)
The inverse of (0,2.5) is (2.5,0)
The inverse of (-1,1) is (1, -1)
The inverse of (-2,-1.5) is (-1.5,-2)
This enables us to see that aninverse is a reflection acrossthe line y = x.
Let f(x) be the red function. Let g(x) be the green function.
f(1)=4 g(4)=1
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Here are two parabolas; each one looks like the inverse of the other one. But is the green one a function?
No.
So it is not a true inverse.
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Express the relation shown in the mapping as a set of ordered pairs. Then write the inverse of the relation.
Answer: {(5, 1), (7, 2), (4, –9), (0, 2)}
Relation Notice that both 7 and 0 in the domain are paired with 2 in the range.
Inverse Exchange X and Y in each ordered pair to write the inverse relation.
Answer: {(1, 5), (2, 7), (–9, 4), (2, 0)}
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Express the relation shown in the mapping as a set of ordered pairs. Then write the inverse of the relation.
Answer: Relation: {(3, 2), (–4, 1), (5, 2)}Inverse: {(2, 3), (1, –4), (2, 5)}
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FINDING EQUATIONS OF INVERSES FROM OTHER EQUATIONS
( ) 2 3f x x x is multiplied by 2 and then 3 is subtracted.The inverse of that is add 3 and divide by 2.
1 3( )
2
xf x
Do not confuse the -1 in 1( )f x with a negative
exponent. It just represents the inverse of a function.
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FINDING EQUATIONS OF INVERSES FROM OTHER EQUATIONS
( ) 2 3f x x An algebraic method of finding the inverse:1. Interchange x and y.
2 3y x OR
2 3x y 2. Solve for y.
2 3
3 2
3
2
x y
x y
xy
3. Replace y with1( )f x
1 3( )
2
xf x
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Classwork:26-34even/520
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Get ready for a “Small Quiz” to be written
on your grade sheet.
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TheEnd
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Quiz. Copy the problems and write the answer.
Put your grade paper on the front of your row, quiz side down.
2If ( ) , ( ) 3, ( ) 5 ,
1. find (4) .
2. Find ( ) .
f x x g x x h x x
g h
f h x