Download Ch 6 Notes

Chapter 6
FITZGERALD
MAT 150/51
Given two functions f and g, the composite function, denoted by f  g (read as “f composed with g”
or “f of g”), is defined by  f  g x   f g x . The domain of f  g is the set of all numbers x in
the domain of g such that g(x) is in the domain of f.
Suppose f x  
1
4
and g x  
.
x3
x 1
Find the domain of f  g .
Find f  g .
Find g  f  4
Suppose hx   2 x 2  3 and j x   4 x .
Find (a). h  j 1
(e). h  j x 
(b).
 j  h1
(c). h  h 2
(f).
 j  hx 
(d).
 j  j  1
Chapter 6
FITZGERALD
MAT 150/51
Find functions f and g such that f  g = H if H  x 3  5 .


5
Find functions f and g such that f  g = H if H  x 2  1 .
Sect 6.2
ONE TO ONE FUNCTIONS.
A function is one-to-one if any two different inputs in the domain correspond to two different
outputs in the range. That is, if x1 and x 2 are two different inputs of a functions f, then f is one-toone if f x1   f x2  .
One-to-one functions must pass both the Vertical Line Test and the Horizontal Line Test.
f x   x 2
f x   x 3
Chapter 6
INVERSE FUNCTIONS
FITZGERALD
f
1
MAT 150/51
x  is read as “the inverse of f x  ”
A Relation such that every x coordinate in the domain corresponds to exactly one y coordinate in the
range and every y coordinate in the range corresponds to exactly one x coordinate in the domain.
Domain of f  Range of f
1
Range of f  Domain of f
Verifying functions are inverses. Suppose f x   2 x  3 and g x  
Verifying functions are inverses. Suppose f x  

1
x  3
2
1
1
and g x    1
x 1
x
Find the inverse of f x  .
f x    3,27,  2,8,  1,1, 0,0, 1,1, 2,8, 3,27
Graph the inverse.
f x 

f 1  f x   x  f f 1 x 
1
Find the inverse of f x  . f x   23 x  5  1
Chapter 6
FITZGERALD
Find the inverse of f x  . f x   3 x  4  5 .
Find the inverse of f x  . f x    x  5  3 .
Find the inverse of f x  . f x   2x  1  7 , x  1
2
MAT 150/51
Chapter 6
Sect 6.3
FITZGERALD
MAT 150/51
EXPONENTIAL FUNCTIONS f  x   b where b  0 and b  1
x
f  x   y  3x
x
y
-1
0
1
2
Common ratio
Five characteristics of basic exponential functions.
1. ____________________________________________________
2. ____________________________________________________
3. ____________________________________________________
4. ____________________________________________________
5. ____________________________________________________
TRANSFORMATIONS
Given any exponential function,
f  x  a  r
bx c 
d
a  ____________________________________________________
b  ____________________________________________________
bx  c  ________________________________________________
d  ____________________________________________________
Chapter 6
FITZGERALD
MAT 150/51
Translate f  x   3x
Graph f  x   3x
Graph f  x   3x 2
Graph f  x   3x  5
Graph f  x   31 x
Graph f  x   3x 2  1
Graph f  x   3 x
SOLVING EXPONENTIAL EQUATIONS (one to one bases)
Solve for x.
53 x  4  5 x
4 x 1 
1
32
2 2 x 7  8
4x  2x  8
2
27 x 2  9
Chapter 6
FITZGERALD
If 5 x  2 , then find 5 3 x .
Find the exponential function from the graph.
MAT 150/51
If 3 x  4 , then find 9  x .
Chapter 6
Sect 6.4
FITZGERALD
MAT 150/51
INVERSE OF EXPONENTIAL FUNCTIONS f  x   b where b  0 and b  1
x
f  x   y  3x
x
Switch x and y. x  3
y
y
-1
0
1
2
x  3y is now given a special function name…. f  x   y  log3  x  . Read as “log base 3 of x”
Five characteristics of basic logarithmic functions.
1. ____________________________________________________
2. ____________________________________________________
3. ____________________________________________________
4. ____________________________________________________
5. ____________________________________________________
TRANSFORMATIONS
Given any exponential function,
f  x   a  log r  bx  c   d
a  ____________________________________________________
b  ____________________________________________________
bx  c  ________________________________________________
d  ____________________________________________________
Chapter 6
Translate f  x   log3  x 
FITZGERALD
MAT 150/51
Graph f  x    log3  x 
Graph f  x   log3  x  2 
Graph f  x   log3  x   5
Graph f  x   log3 1  x 
Graph f  x   log3  x  2   1
Graph f  x    log3   x 
EQUIVALENT EQUATIONS for LOG. & EXP. y  logb  x  converts to b y  x
Exponential to logarithmic.
53  125
Logarithmic to exponential.
log7  49   2
23 
1
8
log 4  32   2.5
m  11
log
7
7  2
Chapter 6
FITZGERALD
NATURAL BASE
y  e x , the inverse function is y  ln  x  .
MAT 150/51
BASE 10
y  10 x , the inverse function is y  log  x  .
Find the inverse.
f  x   3  log  x  5  1
f  x   4  5
x2
1
Solve the equations.
log3  4 x  7   2
Sect 6.5
log x  64   2
3e2 x  30
Chapter 6
FITZGERALD
EXPONENTIAL PROPERTIES
LOGARITHM PROPERTIES
logb 1  0
b0  1
logb  b   1
b1  b
b
MAT 150/51
logb  M 
logb  b M   M
M
bm  bn  bmn
log b  MN   log b  M   log b  N 
bm
mn

b
bn
M
log b 
N
b 
m n

  logb  M   logb  N 

log b  M n   n logb  M 
 b mn
ONE-TO-ONE BASE PROPERTIES
logb  m   logb  n   m  n
bm  bn  m  n
CHANGE-OF-BASE FORMULA
logb  M  
log  M  ln  M 

log  b 
ln  b 
Use the above properties to find the exact value of each expression. No Calculator!
 
log 2 2
e
log
e2
3
 9
11
e
ln 13
log3 8  log8  9 
log 2  48  log 2  3
log6  9   log 6  4 
Chapter 6
FITZGERALD
MAT 150/51
Suppose logb  2   0.6931 , logb  3  1.0986 , and logb  5  1.6094 . Use the properties of logs to
evaluate the expression.
logb 12 
logb  2.7 
logb  30b 
logb 6
Use your calculator and Change-of-Base formula to find the following to three decimals.
log13  39 
log 21  5
log
Expand the expression as the sum and/or difference of single logarithms.
log 2 8x 2 y 
 4 y 
log 3 

 9  x  13 


Condense to a single logarithm.
1
4  log5  x   2  3  log5  y    log5 11
2
log3  x3  1  log3  x 2  1
11
8
Chapter 6
FITZGERALD
Sect 6.6 SOLVING EXPONENTIAL AND LOG EQUATIONS.
logb  m   logb  n   m  n
Solve.
log3  3x  1  log3  2 
vs
MAT 150/51
logb  x   y  b y  x
log3  3x  1  2
2  log5  x   3  log5  4 
3  log 2  x  1  log 2  4   5
log 2  x  7   log 2  x  8  1
log 4  x 2  9   log 4  x  3  3
Chapter 6
FITZGERALD
3x  1  14
 x 1
2
22 x  2x 12  0
32 x  3
x 1
4  0
Use a graphing calculator to solve.
e x  ln  x   4
Sect 6.7
e2 x  x  2
12 x 
5
MAT 150/51
Chapter 6
FITZGERALD
FINANCIAL MODELS with EXPONENTIAL FORMULAS
1.
2.
3.
4.
MAT 150/51
Determine the Future value of an investment.
Calculate the Effective Rate (or APR)
Determine the Present Value of an investment.
Determine the rate or time to Double an investment.
FORMULAS
Simple Interest Formula:
I  P  r t
I = Interest earned, P = Principal, r = rate of annual interest as a decimal, t = time in years
Note: If the interest rate is defined by a different time unit then time is defined as the same.
Example. A monthly interest rate of 0.75% means that time is measured in months.
Compound Interest Formula:
r

A  P  1  
n

nt
A = Accounts future value and n = number of compounds in one year.
A  Pert
Continuous Compounding Formula:
Effective Rate of Interest Formulas:
re
n
Compounding “n” time per year:
Compounding Continuously:
r

re  1    1
n

re  er  1
Present Value Formulas:
r

P

A

1



Compounding “n” time per year:
n

 rt
Compounding Continuously: P  Ae
 nt
Chapter 6
FITZGERALD
MAT 150/51
$500 is borrowed for 6 months at annual rate of 6%. What is the interest owed?
$8000 is invested at 5% for 3 years. What is the future value of the account for the following
compounds…
…quarterly?
… monthly?
… daily?
… continuously?
What is the effective rate of interest (or APR) 6% compounded ….
… monthly?
… continuously?
What rate of interest compounded quarterly will yield an effective interest rate of 14%?
$8000 is invested at 5% compounded monthly. How long will it take to double the investment?
Bacteria #123 grows at a rate of 9% continuously. How long until the population triples?
Your need $10,000 for a car in 4 years. What is the minimum investment needed for a CD account
that pays 3.5% quarterly for 4 years?
Chapter 6
Sect 6.8
FITZGERALD
MAT 150/51
Exponential Growth and Decay.
A  Pert
kt
Authors have a tendency to change the variables. A  t   A0 e
We use the continuous compounding formula.
Growth is when k > 0 and Decay is when k < 0.
NEWTON’S Law of Cooling.
u  t   T   u0  T  ekt
k<0
T = constant Temperature of the surrounding medium.
u0 = initial temperature of the heated object.
t = time
A colony of bacteria grows continuously at 4.5%. The colony starts with 100 grams. Write a
function for the growth where N is measured in grams and t is measured in days
Graph the function on your calculator with the following window; 0< t < 30 and 0 < N(t) < 400.
What is the population in 5 days?
How long will it take for the population to reach 150grams?
How long will it take for the population to double?
Chapter 6
FITZGERALD
MAT 150/51
An object is heated to 100oC and is then allowed to cool in a room whose air temperature is 30oC. If
the temperature of the object is 80oC after 5 minutes, when will the temperature be 50oC?
Graph the function from above in the window 0< t < 60 and 0 < u(t) < 120 Verify the answer.
What is the elapsed time before the object is 35oC? (use the calculator)
What is happening to the temperature as time passes?
Chapter 6
FITZGERALD
Sect 6.9
BUILDING EXPONENTIAL AND LOG functions from data.
Your rich uncle left you some cash in a mutual fund that
will mature in 10 years, only if you can create an
exponential function that matches the data provided.
The function must be in the form A = Pert.
How much is the mutual fund worth when it matures in 10
years?
The following data represents the price and quantity
supplied in 2009 for Dell Computers.
Build a logarithmic function in the form y  a  b  ln  x  .
How many computers were supplied when the cost was
$1650?
MAT 150/51
Year, x
Account value y
0
1
2
3
4
5
6
20000
21516
23355
24885
27484
30053
32622
Price, x
Quantity supplied y
2300
2000
1700
1500
1300
1200
1000
180
173
160
150
137
130
113