Section 5.1: Inverse Functions
1. Let h = {(−8, 5), (4, −2), (−7, 1), (3.8, 6.2)} what is h inverse?
Let g = {(2, 4), (−1, 3), (−2, 0)} what is the inverse relation?
Let y = x2 − 5x, find the equation for the inverse of the relation.
2. One to One: (1-1) If different inputs have different outputs (If a 6= b then f (a) 6= f (b) or if
f (a) = f (b) then a = b.)
Year Cost of Postage
(in cents)
x
y
1996
32
−3 −27
1998
−2 −8
2000
33
−1 −1
2002
37
0
0
2004
1
1
2006
39
2
8
2007
41
3
27
2008
42
2009
44
Horizontal Line Test:
Drawing One to One Functions/Relations: (graph of f −1 is the reflection of f across the
line y = x.)
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3. Facts/Notation:
f −1 (x) is f -inverse of x
f −1 (x) 6=
1
f (x)
If a function f is one to one then f −1 is a function
The domain of a 1-1 function is the range of f −1 .
The range of a 1-1 function is the domain of f −1 .
A function that is increasing over its entire domain or is decreasing over its entire domain is 1-1.
1-1 functions:
(a)
(b)
(c)
Non 1-1 functions:
(a)
(b)
(c)
Finding formulas for Inverses
(a) Replace f (x) with y
(b) interchange x and y
(c) Solve for y
(d) Replace y with f −1 (x).
If f is a 1-1 function then f −1 is the unique function such that
(f −1 ◦ f )(x) = x for all x in the domain of f .
(f ◦ f −1 )(x) = x for all x in the domain of f −1 .
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4. Find the inverse of f (x) = 2x − 3 and graph the two.
5. Find the inverse of g(x) = x3 + 2
6. Find the inverse of h(x) =
3x + 5
x−2
7. Restricting domain: y = x2 − 2 is not one to one, but if we let x ≥ 0 then the function is one to
one and we can find the inverse function.
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Section 5.2: Exponential Functions and Graphs
1. Exponential Function f (x) = ax for x a real number and a > 0, a 6= 1. Note a is called the base.
Examples of Exponential Functions:
Non-Examples of exponential functions:
2. Graph the following functions:
(a) 2x
x
1
(b)
2
(c) 2x−2
1
(d) 2x − 4
(e) 5 − 0.5x
3. Compound Interest: (Formula will be provided, list of variables will not)
r nt
A=P 1+
n
A-future value
P -principal
t-years
r-interest rate (in percent)
n-compound period
Continuous Compound Interest:
A = P ert
e ≈ 2.718
4. Write a function if $100, 000 is invested at 6.5% interest, compounded semi-annually. How much
in 10 years?
2
5. Graph the following:
(a) ex and 2x
(b) ex+3
(c) e−0.5x
(d) 1 − e−2x
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Section 5.3: Logarithmic Functions and Graphs
1. Graph y = 2x and the inverse.
2.
y = loga x
is a number y such that x = ay where x > 0, a > 0 and a 6= 1.
Use the above information to answer the following:
(a) log2 8 =
(b) log3 9 =
(c) log 10, 000 =
(d) log 0.01 =
A few properties (will repeat in next section as well):
loga 1 = 0
loga a = 1
log10 x = log x
loge x = ln x
logb M =
loga M
for all a, b and M > 0.
loga b
1
3. Convert the following to a logarithm:
(a) 16 = 2x
(b) 10−3 = 0.001
(c) et = 70
4. Convert to an exponential
(a) log2 32 = 5
(b) loga Q = 8
(c) logt M = x
5. Use calculator to solve (round to two decimals as needed):
(a) log 645, 778
(d) ln 645, 778
(b) log 0.000 023 9
(e) ln e
(c) log(−3)
(f) ln 1
(g) Find log5 8 using common logarithm.
6. Graph the following:
(a) ln(x + 3)
2
(b) 3 − 12 ln x
(c) | ln(x − 1)|
7. Earthquake Intensity, I, measured on a Richter Scale of magnitude R is
R = log
I
I0
where I0 is the minimum intensity to be measured. On March 11, 2011 there was an earthquake of
intensity of 109.0 · I0 off the coast of Japan. What is the magnitude of this earthquake?
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8. On March 19, 2017 at 6:16AM about 24km from Ponca City, OK there was an earthquake of
magnitude 2.6 what was the intensity of this earthquake?
9. Average walking speed, w in feet per second of a person living in a city of population P , in
thousands, is given by
w(P ) = 0.37 ln P + 0.05.
(a) Population in Savannah, Georgia is 136, 286 find average walking speed.
(b) Population in Philadelphia is 1, 526, 006 find the average walking speed.
(c) The average walking speed of a city is about 2.0 ft/sec, what is the approximate population of
the city.
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Section 5.4: Properties of Logarithmic Functions
Properties of Logarithms: Assume A and B are positive real numbers, a, b > 0, a, b 6= 1 and
p is any real number.
Properties of Logarithms:
• loga (A · B) = loga A + loga B
• loga (A/B) = loga A − loga B
• loga Ap = p loga A
loga M
• logb M =
loga b
• loga a = 1
• loga 1 = 0
• aloga x = x
• loga ax = x
1. Expand, using the properties of logarithms:
(a) log
xy
z
r
(b) log
2x − 1
x+1
(c) log
x
yz
(d) log
(x2 y − 1)
y3z
1
√
(e) ln xy
(f) ln
3x2
y3
(g) ln ((x + y)2 (x − y))
√
(h) ln
x+2
x(x − 1)
2. Contract, expressing the answer as a single logarithm:
(a)
1
(log x − log(x + 1))
3
(b) ln x + ln(x − 1)
2
(c) ln(x + 1) − ln x
(d) 2 ln x − 3 ln y
(e)
1
ln(x + y)
2
(f) ln x − 2 ln(2x − 1)
(g) log(x2 + x − 30) − log(x2 − 36)
3. Given loga 2 ≈ 0.301 and loga 3 ≈ 0.477 find each of the following, if possible.
(a) loga 6
3
(b) loga
2
3
(c) loga 81
(d) loga
1
4
(e) loga 5
(f)
loga 3
loga 2
4. Simplify the following:
(a) loga a8
(d) 4log4 k
(b) ln e−t
(e) eln 5
(c) log 103k
(f) 10log 7t
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Section 5.5: Solving Exponential Equations and Logarithmic Equations (I think this is
the entire point of talking about Exponential and Logarithmic Functions!)
1.
ax = ay implies x = y for a > 0, a 6= 1
loga M = loga N implies M = N for M, N > 0, a > 0, a 6= 1
2. Solve:
(a) 25x = 220x+10
(b) 23x−7 = 32
(c) 53x+2 · 25x = 125x
2
(d) 2x+1 = e5
1
(e) 2x+1 = 52x
(f) 2 + 4n = 7
(g) 10t = 3
(h) 5 · 10t = 3
2
3. Solve: (Note, you must check logs!)
(a) log3 x = −2
(b) logx + log(x + 3) = 1
(c) log3 (2x − 1) − log3 (x − 4) = 2
(d) ln(4x + 6) − ln(x + 5) = ln x
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Section 5.6: Applications and Models: Growth and Decay; Compound Interest
Exponential growth and decay functions can be written in the form
P (t) = P0 ekt .
If k > 0 this is growth, if k < 0 this is decay. You will be expected to know this! Draw a graph:
1. In 2011, the population of Mexico was about 11.3 million, and the exponential growth rate was
1.1% per year.
(a) Find and graph the exponential growth function.
(b) Estimate the population in 2015.
(c) After how long will the population be double what it was in 2011?
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Compound Interest:
r nt
A=P 1+
n
Interest Compounded Continuously: (note book writes in form P (t) = P0 ekt to show similarity
to exponential growth function.)
A = P ert
2. Suppose that $2000 is invested at interest rate k, compounded continuously, and grows to 2504.65
in 5 years.
(a) What is the interest rate?
(b) Write the exponential growth function with the above interest rate and above initial value.
(c) What will the balance be after 10 years? (Is this above or below the amount given in the initial
problem? Use this information to make sure that you are in the ballpark.)
(d) After how long with the $2000 have doubled?
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Doubling Time: Book gives formula, do not memorize, just use the exponential function
P (t) = P0 ekt
(If you have P0 = 100 items and want this to double what will P (t) be? What about P0 = 567? For
each problem, after you complete your first step to solve for t what are you left with?) The moral of
the story is:
3. In May 2010, a 1932 painting, Nude, Green Leaves, and Bust, by Pablo Picasso, sold at a New
York City art auction for $106.5 million to an anonymous buyer. This is a record price for any work
of art sold at an auction. The painting had belonged to the estate of Sydney and Francis Brody, who
bought it for $17, 000 in 1952 from a New York art dealer, who had acquired it from Picasso in 1936.
(Sources: Associated Press, ”Picasso Painting Fetches World Record $106.5M at NYC Auction,” by
Ula Ilnytzky, May 4, 2010; Online Associated News papers, May 6, 2010)
(a) Write an exponential function which describes this situation, P (t) where P is price and t is
years since 1952.
(b) What would the value be this year?
(c) What is the doubling time for the value of the painting?
(d) What year will the value of the painting be $240 million, according to your function?
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Logistic Function:
a
1 + be−kt
y = a is a horizontal asymptote of the graph (this is called the limiting value). What is the y-intercept?
P (t) =
4. A lake is stocked with 400 fish of a new variety. The size of the lake, the availability of food, and
the number of other fish restrict the growth of that type of fish in the lake to a limiting value of 2500.
the population gets closer and closer to this limiting value, but never reaches it. The population of
fish in the lake after time t, in months is given by the logistic function:
P (t) =
2500
1 + 5.25e−0.32t
(a) Graph P (t)
(b) Find the population after 0, 1, 5, 10, 20 months.
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Exponential Decay: Be careful with the sign of the power and the description.
P (t) = P0 e−kt where k > 0
is the same thing as
P (t) = P0 ekt where k < 0.
(given k is the same value in magnitude in both situations.) Your book uses the first function. k is
the decay rate.
As in exponential growth we can talk about doubling time, in exponential decay we can talk about
half-life.
5. Bismuth (Bi-210) has a half-life of 5 days. Suppose there is an initial value of radioactive bismuth
of P0 write a function P (t) where t is days and P is the amount of bismuth.
6. The radioactive element carbon-14 has a half-life of 5750 years. The percentage of carbon 14
present in the remains of organic matter can be used to determine the age of that organic matter.
Archaeologists discovered that the linen wrapping from one of the Dead Sea Scrolls had lost 22.3% of
its carbon-14 at the time it was found. How old was the linen wrapping?
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