COLLEGE ALGEBRA PRACTICE FINAL (REVISED FALL, 2013)
I.
Sketch the following graphs:
1.
3y  x  4  0
2.
f  x    x  1  4
3.
g  x    x2  8x  17
4.
y  x2
5.
y  x 2
6.
f  x 
7.
f ( x) 
8.
f ( x)  ( x  2)( x  1)( x  3) 2
9.
y   x  x  2 x  2
10.
f  x   3x
11.
g  x   3 x
12.
g  x   3x  2
13.
g  x   3x  2
14.
f ( x)  log3 x
15.
f  x   log  x  2
16.
x  4 x  y  6 y  4
17.
 1, if x  0

f ( x)   x  1, if 0  x  2
 2, if x  2

x2  6 x  8
x2  x  2
2
2
2
1
x3
Use the graph of f  x  to sketch the following graphs:
5
4
18.
g  x   f  x  2
19.
h  x   f  x  3
20.
j  x   f  x  1  3
3
2
1
-5 -4 -3 -2 -1
1
2
3
-1
4
5
-2
-3
-4
-5
II.
Find all Solutions:
21.
x2  4 x  3
24.
27.
22.
1  13  x  x
23.
2x 3  3x 3  2  0
25.
36
2x
1 
x3
x2  9
26.
3 x  2  5  10
3x  7  2  9
28.
x3  5x2  6x
29.
x2  4 x  3
0
x2
2
1
III.
Functions, Polynomial Functions, and Rational Functions
30.
Find the equation of the line in slope intercept form
(i)
through the points  3, 4  and  6, 7  .
(ii)
31.
Find the coordinates of the vertex for f  x    x2  4x  3 .
32.
Determine the asymptotes of the graph of f  x  
7  3x 2
x2  5x  6
1
3x  7  x  6  3
through (-9, 2) and perpendicular to 9x  12 y  24
33.
34.
35.
List the possible rational zeros for p  x   2x4  9x3  4x2  21x  18
Find all zeros (real and non-real).
f ( x)  3 x 3  2 x 2  2 x  1
(i)
For the given functions, find
(i)
f  x  h  f  x
f  x   4x  3
p  x   2x4  9x3  4x2  21x  18
(ii)
h
, h  0.
(ii)
f ( x)  x 2  3 x
36.
Find a polynomial of least degree, with integer coefficients and the given zeros. Write answers in expanded form.
(i)
(ii)
7, 5i
1 , 2i
37.
For the given functions f and g, find f g and g f
(i)
38.
40.
f ( x) 
x
x 1
2
(ii)
f ( x)  3x 2  5 x  2
2
x 1
f ( x) 
(iii)
(iii)
f–g
1  5x
x2
(iv) f ( x)  log( x  3)
(iv) f / g
f is a one-to-one function, find f –1.
f (x) = (x + 2)2 , x > – 2
(ii)
f ( x) 
x3
5x  2
(iii)
f ( x)  10  x  2 
For each graph, find the domain, the range, and whether it is a function.
(i)
42.
f ( x)  x  1; g ( x) 
If f ( x)  x and g(x) = 3x – 5, find the following function
(i)
f+g
(ii)
fg
(i)
41.
(ii)
Find the domain of each function.
(i)
39.
f ( x)  3x  1; g ( x)  x 2
(ii)
Verify that the functions f and g are inverses of each other by showing that
 f g  x   x and  g f  x   x
(i)
1
f (x) = 3x + 4; g ( x)  ( x  4)
3
IV.
Logs and Exponentials
43.
Solve to 2 decimal places: 438  200e0.25 x
44.
Find the solution set for:
(i)
log4  x  1  2  log4 3x  2
(ii)
(ii)
2
f (x) = x3 – 8; g ( x)  3 x  8
log x  2  log  x  48
3
45.
An investment of $5000 is compounded continuously for 20 years. What interest rate would yield $16,600? (Round to
the nearest percent.)
46.
On her 18th birthday, Jill invests $25000 in a trust fund that pays 8% interest compounded continuously. How old is she
when the account is worth a million dollars?
47.
How long would it take an $8000 investment to double in value at 4% interest compounded semiannually (twice a year)?
48.
49.
The population of a town increases according to the model P t   2500e0.0293t , where t is the time in years, with t  0
corresponding to 2005.
(i)
Estimate the population in the year 2015.
(ii)
Estimate when the population will reach 4300.
Evaluate.
(i)
ln e3
(ii)
log5 7 (approx)
50.
Expand as much as possible. log b
V.
Matrices and Systems
51.
52.
53.
54.
55.
5
(iii)
log3 1
(iv)
log3  log3 27 
x2
y2 z
 4 5 
For the matrices A  3 2 and B  
:
 1 0 
(i)
Find AB
(ii)
Find B  B
(iii)
Find the determinant of B
1 5 
Find the inverse of 
.
1 6 
Solve the systems of non-linear equations
 x 2  y 2  25
(i)
(ii).

2 x  4 y  10
 xy  2

3 x  y  5
Solve the systems of linear equations
 x  y  2 z  19

(i)
(ii)
3x  y  2 z  31
 x  3 y  2 z  25

 x  y  3z  8

 3x  y  2 z  2
2 x  4 y  z  0

Graph the solutions to the systems
(i).
 y  x 2  0

 x  y  3
(ii)
6 x  10 y  60
 6 x  3 y  24


x0


y0
VI.
Series, Sequences, and the Binomial Theorem
56.
Write the first 4 terms of the sequence whose general term is: an 
57.
Find the third term in the expansion of  2 p  3q 
 2 
n
3n  1
7
59.
 11
What is   ?
7
Write a formula for the nth term of the arithmetic sequence 3 ,  4 , 11 , 18 ,
and identify the 10th term.
60.
Write a formula for the nth term of the geometric sequence 3 , 9 ,  27 , 81 ,
and identify the 10th term.
58.
3
VII.
Applications
61.
A store sells bluegrass seed that is worth $6 per pound and ryegrass seed that is worth $3 per pound. How much of each
should be mixed to obtain 80 pounds of a blend that is worth $5 per pound?
62.
A rectangular painting measures 14 inches by 17 inches and contains a frame of uniform width around the four edges.
The perimeter of the rectangle formed by the painting and its frame is 86 inches. Determine the width of the frame.
63.
The area of a rectangular wall of a barn is 75 square feet. Its length is 10 feet longer than the width. Find the length and
width of the wall of the barn.
64.
A farmer has 600 meters of fencing, and wants to enclose a rectangular field
that borders a river. If he does not fence the side along the river, find the
length and width of the plot that will maximize the area.
65.
A penny is thrown up in the air from a building. Its height in feet after x seconds is given by
h( x)  16 x 2  24 x  75 .
(i) when does it reach its max height?
(ii) what is its max height?
(iii) when does it hit the ground?
66.
When a crew rows with the current, it travels 36 miles in 4 hours. Against the current, the crew rows 20 miles in 4 hours.
Find the rate of rowing in still water and the rate of the current.
VII.
Miscellaneous
67.
What is 3 
68.
Divide:
69.
Sketch the graph of a function with the following characteristics:

(i)
(ii)
(iii)
(iv)
(v)

9  2  5i  ?
3i
4  3i
Exactly one horizontal asymptote at y  3
Exactly two vertical asymptotes at x  3 , x  2
Exactly two zeros at x  1 , x  1
 1
A y  intercept at  0, 
 2
Passes through the indicated points
4
(vi)
COLLEGE ALGEBRA PRACTICE FINAL SOLUTIONS
2.
4
-5
-4
-3
-2
-4
-3
-2
4
3
3
2
2
2
1
1
1
1
2
3
4
-5
5
-4
-3
-2
-1
1
2
3
4
-5
5
-4
-3
-2
-1
-1
-1
-1
-2
-2
-2
-3
-3
-3
-4
-4
-4
-5
-5
5
-5
3.
4
3
-1
4.
5
5
5
1.
x=-3
6.
4
4
3
3
3
2
2
2
1
1
1
1
2
3
4
5
-5
-4
-3
-2
-1
1
2
3
4
5
-5
-4
-3
-2
-1
-1
-1
-1
-2
-2
-2
-3
-3
-3
-4
-4
-4
-5
-5
-5
7.
3
4
5
1
2
3
4
y=0
5
5
4
-1
2
-5
5
5.
1
8.
5
9.
4
3
2
1
-5
-4
-3
-2
-1
1
2
3
4
5
-1
-2
-3
-4
-5
5
10.
-5
-4
-3
-2
5
11.
5
12.
4
4
4
3
3
3
2
2
2
1
1
1
-1
1
2
3
4
y=0
5
y=0
-5
-4
-3
-2
-1
1
2
3
4
5
-5
-4
-3
-2
-1
1
-1
-1
-1
-2
-2
-2
-3
-3
-3
-4
-4
-4
-5
-5
-5
5
13.
15.
-3
-2
3
3
2
2
-1
4
5
x=2
1
1
2
3
4
y=0
5
-5
-1
-4
-3
-2
-1
1
-1
-2
-3
3
4
1
-4
2
5
14.
4
-5
y=2
-2
5
-3
-4
-4
-5
-5
2
3
4
5
16.
17.
5
18.
4
3
2
1
-5
-4
-3
-2
-1
1
2
-1
-2
-3
-4
-5
5
4
3
2
1
-5
-4
-3
-2
-1
1
2
3
4
5
-1
-2
-3
-4
-5
19.
20.
21.
x  2  7, 2  7
24.
x  8,
27.
 , 6   
30.
(i)
4 
,
3 
11
x  15
3
4
y  x  14
3
y
(ii)
33.
1
8
1 ,  2 ,  3 ,  6 ,  9 ,  18 , 
69.
22.
x4
23.
x  2
25.
x  9 [-3 is extraneous]
26.
x  3,  7
28.
 1,0   6, 
29.
(, 3]  [1, 2)
31.
(2,1)
32.
VA: x  6, x  1
HA: y  3
1
3
9
, ,
2
2
2
34.
(i)
(ii)
35.
(i)
(ii)
37.
(i) ( f g )( x)  3x2  1 , ( g f )( x)  9x2  6x  1
(ii)
4
2x  3  h
(f
g )( x) 
x 1
, ( g f )( x) 
x 1
2
x 1 1
1 1  i 3
x ,
3
2
3
x 1 , 2 , 3 , 
2
x3  7 x2  25x 175
x3  5x2  9x  5
36.
(i)
(ii)
38.
(i) (, 1)  (1,1)  (1, )
(ii)
 , 2  
(iv) (3, )
6
1 
,
3 

1


(iii)  2, 
5
3
4
5
39.
41.
43.
(i)
x  3x  5
(ii)
3x x  5 x
(ii)
(iii)
x  3x  5
(iii)
40.
(iv)
x
3x  5
(i)
D: (, )
R: (,1] yes
(ii)
D: [1, )
R: (, ) no
42.
x  3.14
1
(i)
(ii)
45.
48.
51.
54.
6%
(i)
3351
(ii)
In the year 2023
(18.5 years after 2005)
(i)
10
15
(ii)
11 20 
 4 5 


(iii)
5
(i)
 6, 3, 5
(ii)
1,  1, 2
4
16
,  1,
5
11
56.
1,
59.
an  7n  10 ; a10  60
52.
f 1 ( x)  2  3

x
10
1

1
(i) f  ( x  4)   3  ( x  4)   4  x , g (3x  4)  [(3x  4)  4]  x
3
3

3

(ii) f
44.
f 1 ( x)  x  2
2x  3
f 1 ( x) 
5x  1
(i)

3
 
x8 
3
x8

3


 8  x , g x3  8 
3
x
3

8 8  x
33
47
x2
x
46.
64
47.
49.
(i)
3
(ii)
(iii)
(iv)
1.21
0
1
 6 5 
 1 1 


50.
53.
17.5 years
1
 2 logb x  2 logb y  logb z 
5
(i)
(3,-4), (-5,0)
(ii)
(1/3, 6), (-2, -1)
55.
(i)
(ii)
57.
6048 p 5 q 2
58.
330
60.
an   3 ; a10  59049
61.
26 32 lbs of ryegrass
n
62.
3 inches
63.
5 ft by 15ft
64.
53 13 lbs of bluegrass
150m by 300m
65.
(i) 0.75 sec, 84 ft, 3.04 sec
66.
7 mph, 2 mph
67.
21  9i
68.
9 13
 i
25 25
69.
See graphing answers above
7