Inverse Functions

12.1

SAT Question:

Let and for all integers x

and y. If , what is the value of ?

2 1

2

xx

3

2

yy

2m m

13.

85

.215

.4

. 5

37.

2

A

B

C

D

E

3(2)2 3

2m

23 1 103 5

2 2

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!

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)

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

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

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.

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)}

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

( ) 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.

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

Classwork:26-34even/520

Get ready for a “Small Quiz” to be written

on your grade sheet.

TheEnd

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