Download Inverse Functions

Inverse Functions
12.1
SAT Question:
3y
x2  1
Let x 
and y 
for all integers x
2
2
and y. If m  2, what is the value of m ?
A.
B.
C.
D.
E.
13
8
5
2
15
4
5
37
2
3(2)
m  2 
3
2
32  1 10
3 
 5
2
2
The inverse of putting on your
socks and then your shoes is
taking off your shoes and then
your socks. The beginning and
end of each is the same: bare
feet!
Start with x (bare feet).
Add 5 and then subtract 5.
What do you have?
x. Therefore adding and
subtracting are inverses.
What is the inverse of
multiplication?
Division. What is the inverse of
squaring?
Square rooting. What is the
inverse of cube root?
Cubing.
The inverse of a set of ordered pairs is
found by exchanging x and y.
Given A =
(2, 4),(6,8),(10,12)
(4, 2),(8,6),(12,10)
The inverse of A = _____________________
Is this a function? Yes
Every function has an inverse, but the inverse is not
necessarily a function. If the inverse is also a
1
f
( x) and is read
function, it is denoted by
“f inverse.”
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 an
inverse is a reflection across
the line y = x.
Let f(x) be the red function. Let g(x) be the green function.
f(1)=4
g(4)=1
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.
Express the relation shown in the mapping as a set of
ordered pairs. Then write the inverse of the relation.
Relation Notice that both 7 and 0
in the domain are paired
with 2 in the range.
Answer: {(5, 1), (7, 2),
(4, –9), (0, 2)}
Inverse Exchange X and Y in each ordered pair to write
the inverse relation.
Answer: {(1, 5), (2, 7), (–9, 4), (2, 0)}
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)}
FINDING EQUATIONS OF INVERSES FROM
OTHER EQUATIONS
f ( x)  2 x  3
x is multiplied by 2 and then 3 is subtracted.
The inverse of that is add 3 and divide by 2.
x3
f ( x) 
2
1
1
f
( x) with a negative
Do not confuse the -1 in
exponent. It just represents the inverse of a function.
FINDING EQUATIONS OF INVERSES FROM
OTHER EQUATIONS
f ( x)  2 x  3
OR
y  2x  3
An algebraic method of finding the inverse:
1. Interchange x and y.
1
3. Replace y with f ( x)
x  2y 3
2. Solve for y.
x  2y 3
x  3  2y
x3
y
2
x3
f ( x) 
2
1
Classwork:
26-34even/520
Get ready for a “Small Quiz”
to be written
on your grade sheet.
The
End
Quiz. Copy the problems and write the
answer.
If f ( x)  x , g ( x)  x  3, h( x)  5 x,
2
1. find g  h(4)  .
2. Find f  h( x)  .
Put your grade paper on the front of
your row, quiz side down.