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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
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
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:
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,
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
It does not even matter what the angle is, we only
need to find:
oppadj
Find the missing side length(s)
o
2 2 21 6o 35o
351 35
Is the result positive or negative?
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
It does not even matter what the angle is, we only
need to find:adjhyp
Find the missing side length(s)
h
2 2 21x h 2 1h x
2
1
1x
Is the result positive or negative?
12 2Since - tan ,
cos 0
x