5.1 Inverse Functions
For f (x)  x  5 , we say “f is the function that adds five to every
input.” The inverse function of f is the function that undoes
“adding five to each input,” that is, “subtracting five from each
input.” The notation for the inverse function of function f is f 1
(read “f-inverse”).
Function
f (x)  x  5
f 1 (x)  x  5
Rule
Add five to each input
Subtract five from each input
g(x)  x  7
g1 (x)  x  7
Subtract seven from each input
Add seven to each input
h(x)  x  5
h 1 (x)  5x
Divide each input by five
Multiply each input by five
Notice that f (x)  x  5 from set A = {1, 2, 3, 4} to set B = {6, 7,
8, 9} can be represented as
f (x)  x  5 : {(1, 6), (2, 7), (3, 8), (4, 9)}, then
f 1 (x)  x  5 : {(6, 1), (7, 2), (8, 3), (9, 4)}
Domain of f
1
Range of f
f (x)  x  5
2
3
4
Range of f-1
6
7
f 1 (x)  x  5
8
9
Domain of f-1
5.1 Inverse Functions (Page 2 of 23)
Example A
Consider the function f (x) , where the input x is
Celsius Fahrenheit
the Celsius temperature and the output F  f (x) is
0
32
20
68
the equivalent Fahrenheit temperature. A table of
40
104
equivalent temperatures is shown. Since f converts
60
140
Celsius temperature to Fahrenheit temperature, f 1
80
176
converts Fahrenheit temperature to Celsius
100
212
temperature. The input-output diagram is shown
here. Find
Celsius
Fahrenheit
o
o
a. f (20) =
f
:
C

F
0
32
1
20
68
f (68) =
b.
c,
f (0) =
f 1 (32) =
f (100) =
f 1 (212) =
40
60
80
100
104
140
176
212
f 1 : o F  oC
input of f
1
output of f
output of f
1
input of f
d. List three ordered pairs on the graph of f.
e. List three ordered pairs on the graph of f f 1
Facts for an Invertible Function and its Inverse Function
1. The domain (input) of f 1 is the range (output) of f, and visa
versa.
2. The statements f (a)  b and f 1 (b)  a are equivalent. In
words this says f sends a to b and f-inverse sends b to a.
3. (a, b) is on the graph of f if and only if (b, a) is on the graph of
f 1 .
5.1 Inverse Functions (Page 3 of 23)
Example 1
Let f be an invertible function, where f (2)  5
a. Find f 1 (5)
b. What is a point (i.e. ordered pair) on the graph of f?
c. What is a point on the graph of f 1 ?
Example 2
Some values for an invertible function f are given in the
table.
a. Find f (3)
b. Find f 1 (9)
c. List three points on the graph of f?
d. List three points on the graph of f 1 ?
x f (x)
0 1
1 3
2 9
3 27
4 81
5.1 Inverse Functions (Page 4 of 23)
Example 3
The graph of invertible function f is shown.
1.
Find f (2)
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2.
Find f 1 (5)
Example 4
x
 1
Let f (x)  16   .
 2
a. Find five input-output values for f 1 .
b. Find f 1 (8) .
5.1 Inverse Functions (Page 5 of 23)
Example 5 Reflection Property of f and f -1
The graphs of f (x)  2 x and y = x are shown.
Sketch the graph of f 1 .
x
-3
-2
-1
0
1
2
3
f(x) = 2x
f
23  0.125 (-3, 0.125)
22  0.25 (-2, 0.25)
(-1, 0.5)
21  0.5
(0, 1)
20  1
(1, 2)
21  2
(2, 4)
22  4
(3, 8)
23  8
f 1
(0.125, -3)
(0.25, -2)
(0.5, -1)
(1, 0)
(2, 1)
(4, 2)
(8, 3)
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Reflection Property of a Function and Its Inverse
For and invertible function f, the graph of f 1 is the reflection of
the graph of f across the line y = x.
5.1 Inverse Functions (Page 6 of 23)
Graphing an Inverse Function
For and invertible function f, the graph of f 1 is constructed by the
following steps:
1. Find several points on the graph of f and sketch the graph of f.
2. For each point (a, b) on f, plot (b, a) on f 1 .
3. Sketch f 1 by drawing the curve through the points from step 2.
Example 6
Graphing an Inverse Function
1
Let f (x)  x  1.
3
a. Sketch the graphs of f and f 1 on the
same set of axes.
y
5
x
f (x)
-6
-3
f
f 1
(-6, -3) (-3, -6)
-5
5
0
3
b. Find an equation for f 1 (x) .
-5
x
5.1 Inverse Functions (Page 7 of 23)
Find the Equation of an Inverse Function
For and invertible function y  f (x) , the equation of f 1 is found
by the following steps:
1. Replace f (x) with y.
2. Interchange x and y.
3. Solve for y.
4. Replace y with f 1 (x) .
Example 7
Find the Equation of an Inverse Function
1
Let f (x)  x  1. Find f 1 (x) . Compare your result with
3
Example 4.
5.1 Inverse Functions (Page 8 of 23)
Example 8
Find the Equation of an Inverse Function
Let F  g(x)  1.8x  32 , Where F  g(x) is the Fahrenheit
temperature corresponding to Celsius temperature x.
a. Find the equation for g1 .
b. Find g(25) and explain its meaning in this application.
c. Find g1 (59) and explain its meaning in this application.
Example 9
Find the Equation of an Inverse Function
Find the inverse of f (x)  2x  3 . Verify your results on your
graphing calculator by graphing f, f 1 and y  x using ZSquare.
5.2 Logarithmic Functions (Page 9 of 23)
5.2 Logarithmic Functions
Example 1
Solve 4 x  36
Solve using the CALC/5:intersect function.
Set
Y1  4 x ,
Y2  36 then solve the system
graphically by intersecting the
two functions.
QuickTime™ and a
Photo - JPEG decompressor
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QuickTime™ and a
Photo - JPEG decompressor
are needed to see t his picture.
Notice that the exponential equation 4 x  36 is asking the question
“what is the exponent on base 4 that gives 36?” In logarithmic
notation the solution to 4 x  36 is described as log 4 (36) , read
“logarithm base 4 of 36,” or simply “log base 4 of 36.” The
number log 4 (36) is the exponent on 4 that gives 36. Remember:
A logarithm is an exponent.
5.2 Logarithmic Functions (Page 10 of 23)
Logarithm
For b  0 , b  1, and a  0 , logb (a) is the exponent of b that gives
a. That is,
logb (a)  k if and only if b k  a
The number b is called the base of the logarithm.
Example 2
Find each logarithm.
1. log6 (36)
2.
log 4 (64)
3.
log2 (32)
4.
 1
log 5  
 25 
5.
 1
log 2  
 8
6.
log 7 ( 7 )
7.
log15 (1)
8.
log6 (6)
5.2 Logarithmic Functions (Page 11 of 23)
The Common Logarithm, b = 10
The common logarithm is the logarithm with base 10. We write
log(a) for log10 (a) . That is,
log(a)  k if and only if 10 k  a
Example 3
Find each logarithm.
1. log(1000)
2.
log(0.000001)
Properties of Logarithms
For b  0 and b  1.
1. logb (b)  1 “a positive number to the first power is itself”
2. logb (1)  0 “a positive number to the zero-th power is one”
Definition of a Logarithmic Function
The logarithmic function, base b, is given by
f (x)  logb (x)
where b  0 , and b  1. The domain of the logarithmic function
is the set of positive real numbers. The range of the logarithmic
function is the set of all real numbers
Example 4
Let f (x)  3x . Find
1. f (4)
2. f 1 (9)
5.2 Logarithmic Functions (Page 12 of 23)
Logarithmic Functions and Exponential Functions are
Inverses of Each Other
For b  0 and b  1.
1. For the exponential function f (x)  b x , f 1 (x)  logb (x) .
2. For the logarithmic function f (x)  logb (x) , f 1 (x)  b x .
Example 5
Find the inverse function of
each function.
Function
g(x)  5 x
h(x)  log8 (x)
 1
f (x)   
 2
x
f (x)  log3 (x)
Example 6
Let f (x)  3x . Find
1. f (4)
3. f (2)
2. f 1 (9)
 1
4. f 1  
 27 
Inverse
Function
5.2 Logarithmic Functions (Page 13 of 23)
Example 7
Sketch the graph of y  log 3 (x) .
Steps to Sketch a Logarithmic Function
1. Set f (x)  3x so that f 1 (x)  log3 (x)
2. Find several points on the graph of f and sketch the graph of f.
3. Interchange the components of each ordered pair to find points
on the graph of f 1 .
4. Sketch the curve that contains the points from step 3.
y  3x
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5.3 Properties of Logarithms (Page 14 of 23)
5.3 Properties of Logarithms
Equivalent Logarithmic and Exponential Equations
For b  0 , b  1, and a  0 ,
logb (a)  k if and only if b k  a .
The equation logb (a)  k is called the logarithmic form of the
equation and b k  a is the equivalent exponential form of the
equation.
Steps To Solve Logarithmic Equations
1. Isolate the logarithmic factor.
2. Write the equation in exponential form.
3. Solve.
Example 1
Solve a Logarithmic Equation
1. Solve for x.
3log4 (x)  9
2. Solve for x (to 3 decimal places). 5.2 log(x)  10.7
3. Solve for x. log2 (3x 1)  5
4. Solve for x (to 3 decimal places).
9 log 3 (x 4 )  18
5.3 Properties of Logarithms (Page 15 of 23)
Example 2
Solve for the Base of a Logarithm
1. Solve for b.
logb (81)  4
2. Solve for b (to 3 decimal places). logb (67)  5
Power Property of Logarithms
For b  0 , b  1, and x  0 ,
logb (x n )  n logb (x)
Steps To Solve Exponential Equations
1. Isolate the exponential factor.
2. Take the logarithm of both sides and apply the Power Property
of Logarithms to bring the exponent in front of the logarithm.
3. Solve.
Example 3
Solving Exponential Equations
1. Solve for x (to 3 decimal places). 2 x  12
 
2. Solve for x (to 3 decimal places). 3 4 x  71
3. Solve for x (to 3 decimal places). 5 3x1  17
5.4 Exponential Modeling (Page 16 of 23)
5.4 Exponential Modeling
Example 1
A person invests $4000 in an account at 7%
interest compounded annually. How long will it
take for the value of the investment to double?
Example 2
The number of female competitors in the
Olympic games has greatly increased during
the past 4 decades. Let f (t) represent the
number of female competitors at t years since
1960.
a. Find the appropriate regression model
(linear or exponential) of the data?
A = A0 (1 + i) t
A = Amount after t years
i = Annual interest rate
A0 = Initial principal
Year
1960
1968
1976
1984
1992
2000
Number of
Female
Competitors
610
781
1251
1620
2710
3906
b. Find f (48) and explain its meaning in this application.
c. Find f 1 (7000) and explain its meaning in this application.
5.4 Exponential Modeling (Page 17 of 23)
Example 3
An archeologist discovers a tool made of wood.
a. If 10% of the wood’s carbon-14 remains, how
old is the tool? The half-life of carbon-14 is
5730 years. Write the base to 5 decimal places.
1
P = 100  
2
t t1/ 2
 1
A  100  
 2
b. Find the decay rate and explain its meaning in this application?
t t1/2
5.5 More Properties of Logarithms (Page 18 of 23)
5.5 More Properties of Logarithms
Product and Quotient Properties of Logarithms
For b  0 , b  1, x  0 and y  0
1. Product Property logb (xy)  logb (x)  logb (y)
 x
2. Quotient Property log b    log b (x)  log b ( y)
 y
3. Power Property
logb (x n )  n logb (x)
Example A
Show log(23)  log(2)  log(3)
Example 1
Simplify Logarithmic Expressions
Simplify by writing each expression as a single logarithm with a
coefficient of 1.
1. log b (x)  log b (x  1)
2. 3log b (x)  log b (6x)
3. logb (6x7 )  logb (3x 2 )
4. log b (x)  3log b (5x)  5log b (2x)
5.5 More Properties of Logarithms (Page 19 of 23)
Example 2
Solve Logarithmic Equations
Solve the equation for the exact solution. Then find the
approximate solution to 4 decimal places.
2 log5 (3x)  4 log5 (2x)  3
Step 1: Combine all logarithms
to a single logarithm
with a coefficient of 1.
Step 2: Write the logarithmic
equation in exponential
form and solve.
Example 3
Solve the equation for the exact solution. Then find the
approximate solution to 4 decimal places.
5log7 (x)  2 log7 (3x)  2
5.5 More Properties of Logarithms (Page 20 of 23)
Change of Base Formula
For a and b positive and not equal to one, and x  0
log a (x) log(x)
log b (x) 

log a (b) log(b)
Example 4
a. Find log 3 (81)
b. Find log 2 (12)
Example 5
Write each as a single logarithm.
log 7 (x)
1.
log 7 (4)
2.
log b (8)
log b (2)
Example 6
Sketch the graph of f (x)  log1/2 (x)
y
5
-5
5
-5
x
5.6 Natural Logarithms (Page 21 of 23)
5.6 Natural Logarithms
The Number e
The number e is an irrational number occurring naturally in
mathematics. It is approximately
e  2.718281828459045...
The Natural Logarithm
1. The natural logarithm is the logarithm with base e. The
notation for the natural logarithm is
loge (a)  ln(a) .
2. For a  0 ,
ln(a)  k is equivalent to ek  a .
Example 1
Use a calculator to find the following values t to three decimal
places.
(a) e
(b) ln(50)
Example 2
Solve each equation. Write the exact solution. Then approximate
the solution to three decimal places.
5e x1  100
a.
b.
ln(x)  4
5.6 Natural Logarithms (Page 22 of 23)
Properties of Natural Logarithms
1.
ln(1)  0
2.
ln(e)  1
3.
ln(x n )  n ln(x)
4.
ln(x)  ln(y)  ln(xy)
5.
 x
ln(x)  ln(y)  ln  
 y
Example 4
Solve. Write the exact solution. Then approximate the solution to
three decimal places.
2(5)x  3  63
5.6 Natural Logarithms (Page 23 of 23)
Example 5
Write 5 ln(x)  3ln(2x) as a single logarithm with a coefficient of
one. Simplify.
Example 6
Solve for the exact solution. Then approximate the solution to
three decimal places.
3ln(4x)  ln(5x)  7