MAT 111 Final Exam
Fall 2006
Name ____________________________________
Work each of the following problems in the space provided. Clearly indicate the final answer.
Partial credit will be given based on the work shown, so be neat and thorough._______________
1. Let z = 3 + 4i. Recalling that z represents
the conjugate of z, simplify the expression
z
so that it is standard form.
z
2. Solve the following system of equations.
(Give exact answers, not decimal approximations.)
3x + 4y = 8
2x – 2y = 4
3. Solve the following inequalities. (Give your
answers in interval notation.)
a.
x  4  3  10
b.
( x  3)( x  4)
0
x2
4. Find the equation of the line that is perpendicular to the line 2 x  3 y  12 and passes
through the point (3,1). (Give your answer
in slope-intercept form.)
5. Solve the following equations. (Give exact
answers, not decimal approximations.)
a.
8. For f ( x)  3x  2 and g ( x)  1  2 x find the
following (and simplify):
a. ( f  g )(4)
b. ( f  f )(5)
c. ( f  g )( x)
d. ( f  g )(4)
9. Determine the quadratic function f whose vertex is (3,4) and passes through the point
(1,10) . (Give your answer in slope-intercept
form.)
10. Given f ( x)  3 x  2
a. State the domain of f .
b. State the range of f .
c. Give the equation for the asymptote for the
graph of f .
d. Sketch the graph of f , showing all asymptotes and intercepts accurately.
x  20  x  10
b. x 4  13x 2  36  0
c. 9 2 x 7  27
6. Solve the following equations. (Any decimal approximations of final answers should
be rounded correctly to 3 decimal places.)
a. log 5 (6 x  11)  3
b. 4 35 x  e 7 x
7. The owner of a coffee shop decides to experiment with a new blend by mixing a
Colombian coffee that sells for $9 per pound
with a lower grade of coffee that sells for $5
per pound. She will make 100 pounds of the
new blend and sell it for $8 per pound. How
many pounds of each kind of coffee are required for the blend?
e. Find the equation of the inverse function
f 1 .
f. State the domain of f 1 .
g. State the range of f 1 .
h. Give the equation for the asymptote for the
graph of f 1 .
11. Find the following for the function
 x 2
if  2  x  1

f ( x)  3
if x  1
x  1
if x  1

a. The domain of f :
b. f (1)
c. f (2)
d. f (2)
e. Sketch the graph of f :
14. For each of the following graphs, state the function which corresponds to it.
5
4
3
2
1
0
-4
-3
-2
-1
0
1
2
0
1
2
-1
-2
f (x) 
6
5
4
3
2
1
12. For each, form a polynomial that meets the
given criteria:
a. Opens down, touches the x axis at the
origin and crosses the x axis at 3 and -3.
(Multiply out your answer.)
b. Polynomial is degree 4. Zeros include 3,
-1, and 9i.
2x 2  x  6
13. For the function R( x)  2
x  7 x  10
a.
b.
c.
d.
e.
f.
g.
Find the domain.
Express R(x) in lowest terms.
Find the y-intercept.
Find the x-intercept(s).
Find any vertical asymptotes.
Find any horizontal asymptotes.
Sketch the graph of R, showing all intercepts and asymptotes accurately.
0
-4
-3
-2
-1
-1
-2
g (x) 
15. Use the properties of logarithms to find the exact value of the following expression:
2 log2 7log2 5
16. Rewrite the following expression as a single
logarithm:
1
log 5 ( x  3)  2 log 5 ( x  1)  log 5 ( x  1)
2
17. Use the change of base formula and a calculator
to evaluate log 7 25 ,. (Round your answer to 3
decimal places.)
18. Find the principal to be invested now in order to
have $10,000 after compounding interest
monthly at 8% for 10 years.
19. A culture of bacteria obeys the law of uninhibited growth. If 300 bacteria are present
initially, and there are 750 after 2 hours,
how long will it take until there are 2000
bacteria? (Round your answer to the nearest
tenth of an hour.)
20. The data set below gives the population (in
thousands) of a small city, and x = 0 corresponds to the year 1900.
x (year)
10
30
50
70
90
y (population in thousands)
43.9
40.3
45.4
55.6
71.3
a. Based on the scatter plot (or the rsquared values), which of the following
relations best describes the data? (Circle
the correct answer.)
linear, y  ax  b
quadratic, y  ax 2  bx  c
exponential, y  ab x
logarithmic, y  a  b ln(x)
b. Find the function of best fit. (Round
values to three decimal places.
c. Use the function from part (b) above to
predict the population in the year 2000.
(Round to the nearest whole number.)
21. A manufacturer’s marketing department has
determined that the revenue R (in dollars)
from selling a certain product as a function
of price p (in dollars) is
R( p)  125 p 2  56,000 p
a. Find the price that should be charged in
order to maximize revenue. (Give answer rounded to the nearest cent.)
b. Find the maximum revenue. (Give answer
rounded to the nearest cent.)