Download Math 122, Practice Questions for Final Note. The final test will be

Math 122, Practice Questions for Final
Note. The final test will be cumulative, with approximately 3 to 5 questions coming from the
material from each of the first 4 tests, and 3-5 on the new material on logarithms (Sections
4.2 – 4.5) for a total of 15 to 20 questions. For practice, you may wish to review the chapter
review problems and chapter practice tests from each of the previous chapters (Chapters 5,6,7
and 8), as well as the tests given this quarter. Included here are some questions over sections
4.1 – 4.5; note questions 2(b),4,5,6,7,8,9 are on the new material. You may be required to
complete a portion of the test without the use of a calculator.
Hints and answers are available at the instructor’s website:
http://faculty.lasierra.edu/~jvanderw/classes/m122s10/m122frs10ans.pdf
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
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graph 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
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 .
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 ).
!
√
x3 z 5
in terms of logb x, logb y and logb z.
6. (a) Write logb
y4
xy
√
(b) Write 4 logb (x3 z 3 ) − 2 logb (z y) + 3 logb
as a single logarithm.
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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. (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?
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