Inverse Functions. Inverse Relations The inverse of a relation is the set of ordered pairs obtained by switching the input with the output of each ordered

Inverse Functions

Inverse RelationsThe inverse of a relation is the set of ordered pairs

obtained by switching the input with the output of each ordered pair in the original relation. (The domain of the original is the range of the inverse; and vice versa)

Ex: and are inverses because their input and output are switched. For instance:

5 2f x x

4

22

2'

5

xf x

4

22

5 2f x x 1 25

xf x

4 22f 1 22 4f

Tables and Graphs of Inverses

y = xLine of Symmetry:

Orginal Inverse

X Y

0 25

2 16

6 4

10 0

14 4

18 16

20 25

X Y

25 0

16 2

4 6

0 10

4 14

16 18

25 20

X Y(0,25)

(2,16)

(6,4)

(10,0)

(14,4)

(18,16)

(20,25)

(4,14)

(4,6)

(0,10)

(16,2)

(16,18)

Switch x and y

Switch x and y

Although transformed, the graphs are identical

Inverse and Compositions

In order for two functions to be inverses:

AND

g xf x

f xg x

One-to-One Functions

A function f(x) is one-to-one on a domain D if, for every value c, the equation f(x) = c has at most one solution for every x in D. Or, for every a and b in D:

unlessf a f b a b

Theorems:

1. A function has an inverse function if and only if it is one-to-one.

2. If f is strictly monotonic (strictly increasing or decreasing) on its entire domain, then it is one-to-one and therefore has an inverse function.

The Horizontal Line Test

If a horizontal line intersects a curve

more than once, it’s inverse is not a

function.

Use the horizontal line test to decide which graphs have an inverse that is a function. Make sure to circle the functions.

The Horizontal Line Test

If a horizontal line intersects a curve

more than once, it’s inverse is not a

function.

Use the horizontal line test to decide which graphs have an inverse that is a function. Make sure to circle the functions.

Example

Without graphing, decide if the function below has an inverse function.

32 6f x x

If f is strictly monotonic (strictly increasing or decreasing) on its entire domain, then it is one-to-one and therefore has an inverse function. See

if the derivative is always one sign:

2' 6f x x

Since the derivative is always negative, the inverse of f is a function.

Find the Inverse of a Function

1. Switch the x and y of the function whose inverse you desire.

2. Solve for y to get the Inverse function

3. Make sure that the domains and ranges of your inverse and original function match up.

ExampleFind the inverse of the following: 4 3d x x

4 3x y

3 4x y

23

4

xy

3

4

xy

2

1 3

4

xd x

when 3x

Make sure to check with a table and graph on the calculator.

Switch x and y

Really y =

Solve for y

Restrict the Domain!

Full Parabola (too much)

Only Half Parabola

x=3

Logarithms v Exponentials

The logarithm base a of b is the exponent you put on a to get b:

i.e. Logs give you exponents!

The logarithm to the base e, denoted ln x, is called the natural logarithm.

Definition of Logarithm

if and only

lo

if

ga

x

b x

a b

a > 0

and

b > 0

Logarithm and Exponential Forms

5 = log2(32)

25 = 32

Logarithm Form

Exponential Form

Base Stays the

BaseLogs Give you

Exponents

Input Becomes

Output

Examples

Write each equation in exponential form

1.log125(25) = 2/3

2.Log8(x) = 1/3

Write each equation in logarithmic form

3.If 64 = 43

4.If 1/27 = 3x

1252/3 = 25

81/3 = x

log4(64) = 3

Log3(1/27) = x

ExampleComplete the table if a is a positive real number and:

Domain

Range

Continuous?

One-to-One?

Concavity

Left End Behavior

Right End Behavior

xf x a f x 1f x xa loga x

All Reals

All Positive Reals

All Positive Reals

All Reals

Yes Yes

Yes Yes

Always Up Always Down

lim 0x

xa

lim x

xa

0

lim logax

x

lim logax

x

The Change of Base Formula

loglog

logc

bc

aa

b

For a and b greater than 0 AND b≠1.

The following formula allows you to evaluate any valid logarithm statement:

Example: Evaluate

1.04log 2

ln 2

ln 1.04 17.673

Solving Equations with theChange of Base Formula

2 3.46 1 909x

2 3.46 908x

3.46 454x

3.46log 454x

Solve:

Isolate the base and power

log 454

log 3.46x

4.9289x

Change the exponential equation to an logarithm equation

Use the Change of Base Formula

Properties of LogarithmsFor a>0, b>0, m>0, m≠1, and any real number n.

Logarithm of 1:

Logarithm of the base:

Power Property:

Product Property:

Quotient Property:

log log logm m ma b a b

log log logam m mb a b

log lognm ma n a

log 1 0m

log 1m m

Example 1

3105 7log x

35 5log 10 log 7x

1 35 5 5log log 10 log 7x

15 5 53 log log 10 log 7x

Condense the expression:

Example 2

ln 3 ln 2lnx y

2ln 3 ln lnx y

2ln 3xyExpand the expression:

Example 3

32x

32 4x

4log 2 3x

4 4log 2 log 3x Solve the equation:

AP RemindersDo not forget the following relationships:

ln xe x

ln xe xa b a be e e

aa b

b

ee

e

Inverse Trigonometry

Trigonometric FunctionsS

ine

Cos

ine

Tang

entC

osecantS

ecantC

otangent

Each one of these trigonometric functions fail the horizontal line test, so they are not one-to-one. Therefore,

there inverses are not functions.

In order for their inverses

to be functions, the domains of

the trigonometric functions are restricted so

that they become one-

to-one.

Sin

eC

osin

eTa

ngen

tC

osecantS

ecantC

otangentTrigonometric Functions with Restricted

Domains

2 2: ,D

: 0,D

2 2: ,D

2 2: ,0 0,D

2 2: 0, ,D

: 0,D

Trigonometric Functions with Restricted Domains

Function Domain Range

f (x) = sin x

f (x) = cos x

f (x) = tan x

f (x) = csc x

f (x) = sec x

f (x) = cot x

2 2,

0,

2 2,

2 2,0 0,

2 20, ,

0,

1,1

1,1

,

,

, 1 1,

, 1 1,

Sin

-1C

os-1

Tan-1

Csc

-1S

ec-1

Cot -1

Inverse Trigonometric Functions

Inverse Trigonometric Functions

Function Domain Range

f (x) = sin-1 x

f (x) = cos-1 x

f (x) = tan-1 x

f (x) = csc-1 x

f (x) = sec-1 x

f (x) = cot-1 x

2 2,

0,

2 2,

2 2,0 0,

2 20, ,

0,

1,1

1,1

,

,

, 1 1,

, 1 1,

Alternate Names/Defintions for Inverse Trigonometric Functions

Familiar Alternate Calculator

f (x) = sin-1 x f (x) = arcsin x f (x) = sin-1 x

f (x) = cos-1 x f (x) = arccos x f (x) = cos-1 x

f (x) = tan-1 x f (x) = arctan x f (x) = tan-1 x

f (x) = csc-1 x f (x) = arccsc x f (x) = sin-1 1/x

f (x) = sec-1 x f (x) = arcsec x f (x) = cos-1 1/x

f (x) = cot-1 x f (x) = arccot x f (x) = -tan-1x+2

Arccot is different because it is always positive but tan can be negative.

Example 1

Evaluate: 1 12sin

This expression asks us to find the angle whose sine is ½.

Remember the range of the inverse of sine is .2 2,

2 6 2

1Since sin and ,

6 2

1 1sin

2 6

Example 2

Evaluate: 1csc 1

This expression asks us to find the angle whose cosecant is -1 (or sine is -1).

Remember the range of the inverse of cosecant is . 2 2,0 0,

2 2Since csc 1 and 0,2

1csc 12

Example 3

Evaluate: 13tan arcsin

The embedded expression asks us to find the angle whose sine is 1/3.

Draw a picture (There are infinite varieties):

tan

31

It does not even matter what the angle is, we only

need to find:

oppadj

Find the missing side length(s)a

2 2 21 3a 8a 2 2

12 2

24

Is the result positive or negative?

13Since arcsin 0,

tan 0

Example 4

Evaluate: 1 16tan cos ( )

The embedded expression asks us to find the angle whose cosine is -1/6.

Draw a picture (There are infinite varieties):

tan

6

1


need to find:

oppadj

Find the missing side length(s)

o

2 2 21 6o 35o

351 35


1 16 2Since cos ,

tan 0

Ignore the negative for

now.

Example 3

Evaluate: 1cos tan x

The embedded expression asks us to find the angle whose tangent is x.

Draw a possible picture (There are infinite varieties):

cos x

1


need to find:adjhyp

Find the missing side length(s)

h

2 2 21x h 2 1h x

2

1

1x


12 2Since - tan ,

cos 0

x

White Board Challenge

1sec 2

4

Evaluate without a calculator:


1 3cot

3

2

3



arccos 2 3x

2x



1cot csc 5

2 6



1tan sin x

21

x

x


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Inverse Functions. Inverse Relations The inverse of a relation is the set of ordered pairs obtained by switching the input with the output of each ordered