Download Math 121, Practice Questions for Final Note. The final test will be

Math 121, Practice Questions for Final
Note. The final test will be cumulative, with approximately one quarter of the questions coming from
Chapter 4. For practice, you may wish to review the practice questions for each of the previous tests,
as well as the tests and quizzes given this quarter. Included here are some questions over Chapter 4,
which is the new material since the last test. See the instructor’s website for finals from other terms.
Note: in all questions b > 0 and b 6= 1 where b is an exponential or logarithmic base.
1. Determine whether the following functions are inverse functions. If they are inverse functions,
sketch the both functions together on the same coordinate axis without using a graphing calculator.
x+5
(a) F (x) = 2x − 5, G(x) =
.
2
(b) p(x) =
x−5
2x
, q(x) =
.
2x
x−5
2. (a) Find the inverse function of f (x) = x5 − 3. Verify that the function you have found is the
inverse function to f .
(b) What is the inverse function of h(x) = log 1 x? Verify that this is the inverse function, and graph
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h(x) and its inverse on the same coordinate axis without using a graphing calculator.
3. Suppose f is a function and g is its inverse function, suppose also the domain of f is [3, 10] and
the range of f is [−2, 20] with f (3) = −2 and f (10) = 20.
(a) Is f one-to-one?
(b) Is g one-to-one?
(c) Find the domain and range of g.
(d) If possible, find: g(3), g(−2), g(10), g(20).
4. Solve each equation without using a calculator:
(a) log3 81 = x
(b) 23x−1 = 32
2
(c) ln eπ = x
(d) b3x+2 = (b4 )x+1
5. Complete this question without the use of a graphing calculator or utility.
(a) Sketch the graph of f (x) = 3|x| .
(b) Write the equation y = log3 x in exponential form.
(c) Sketch the graph of f (x) = log3 x, then use reflections or translations to sketch the graphs of
g(x) = − log3 x, h(x) = − log3 (x − 2), k(x) = 4 − log3 (x − 2).
(d) Sketch the graph of f (x) = log3 x4 .
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Note: in general, if you are given the graph of y = logb x (or of y = bx without being given b, you
should be able to graph y = bx , y = logb xp , y = logb (x − h) + k, y = k + b( x − h) etcetera when
b, h and k are given. Similarly if you are given the graph of y = bx , you should be able to graph
y = logb x, y = logb xp , y = logb (x − h) + k, y = k + b(x−h) etcetera when p, h and k are given. (Also,
you should be able to estimate b given the graph of y = logb x or y = bx ).
√
6. (a) Write logb
x3 z 5
y4
!
in terms of logb x, logb y and logb z.
xy
√
as a single logarithm.
(b) Write 4 logb (x3 z 3 ) − 2 logb (z y) + 3 logb
2
7. (a) Change the equation log 1 8 = −3 to exponential form.
2
(b) Change the exponential equation 211 = 2048 to logarithmic form.
ex − e−x
1
= .
x
−x
e + 2e
2
(b) Solve the equation 2 + log(3x − 1) = log(2x + 1).
8. (a) Solve the equation
9. (Interest Income) Suppose $18,000 is invested at an annual interest rate of 9% compounded daily.
(a) How much will it be worth after 12 years?
(b) How long will it take until the investment is worth $500,000?
10. (Richter Scale) The magnitude of an earthquake of intensity I on the Richter scale is
I
M = log
I0
where I0 is the intensity of a zero-level earthquake.
(a) Find the magnitude to the nearest 0.1 of the 1999 Joshua Tree earthquake that had an intensity
of I = 12, 589, 254I0 .
(b) Find the intensity of the 1999 Taiwan earthquake that measured 7.6 on the Richter scale.
(c) On March 3, an earthquake in Orange County measured 2.9 on the Richter scale. Compare the
intensity of the 1999 Taiwan earthquake with this earthquake. In other words, how many times more
intense was the Taiwan earthquake?
11. (Exponential Growth) The population of bacteria in a vat of potato salad at Bob’s Big Buffet is
modeled by the exponential growth equation P (t) = P0 ekt . At noon there were 1000 bacteria present
and at 12:30pm there were 1300 bacteria present. How many bacteria were present at: (a) 11:30am
when the Buffet opened? (b) 1:30pm when the Buffet closed? (Assume no one ate any potato salad
because of its funny smell). Use t = 0 at noon, time in hours.
12. (Carbon Dating) Carbon-14 has a half-life of 5730 years, and satisfies the exponential-decay
equation A(t) = A0 ekt . Suppose former LSU president Geraty has excavated an ancient farm and
found a pile donkey bones.
(a) Approximately how long ago did the donkeys live if they have 63% of the the original amount of
Carbon-14 remaining?
(b) Same as (a), except that 72% of the original Carbon-14 remains in the bones.
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