MAC 1105/Summer 2006
College Algebra
Final Review(comprehensive)/sample questions
*** To answer all questions in the test, you must need to be the master of everything that I
covered in the syllabus
Test 1 material:
1. Solve the equation:
100  4 p 5 p  6

6
a)
3
4
3
4
1
b)
 
x( x  3) x x  3
c) ( x  2) 2  x 2  4( x  1)
d) (2 x  1) 2  4( x 2  x  1)
4
3
1
e)


x x  6 x( x  6)
2. Two plane leave Los Angeles at the same time. One heads south to San Diego; other heads
north to San Franscisco. The San Franscisco plane flies 50mph faster. In 1/2 hour, the planes are
275 mile apart. What are their speeds?
3. See exercise # 19, 20 (sec. 1.2, page 102)
4. See exercise #12(sec 1.2, page 101)
5. Solve the equation: 5x 2  3( x 2  3)  22
6. Solve the quadratic equation:
 12 x 2  2 x  24  0
7. Solve the quadratic equation: x 2  5 x  3
8. Determine the number of real solutions to the quadratic equation.
a) 3x 2  4 x  2  0
b) x 2  6 x  5  3x  1
9. A rocket is launched from atop a 105-ft cliff with an initial velocity of 158 feet per second.
The height of the rocket t seconds after launch is given by the equation h  16t 2  158t  105 .
Find the time it will take for the rocket to land.(hint. similar to hw. #29, sec 1.5-page 134)
10. see exercise #6, 8(sec. 1.5, page 131).
11. Solve the following equations:
a) x 3  3x 2  4 x  12  0
b) x 4  1  0
c) 2 x 3  9 x 2  9 x  0
d)  4 x 4  12 x 2  16 x 3
MAC 1105 Final Review - M. RAHMAN
12. Solve the equations provided below. Be sure to check your answers.
x 2  5x  13  3
x  14  x  16
a)
b)
13. Solve the following equations:
9
 8
a) x 
x2
x
7

 1
b) 2
x 9 x3
c) x 2  x  7 x  16
14. Solve the inequality and write the solution using interval notation:
a)  2( x  3)  2 x  5
6  3x
2
b)  6 
5
c) 2 x  5  7
d) 7  6 x  3
Test 2 material
1. Solve the equations provided below. Be sure to check your answers.
x 2  5x  13  3
x  14  x  16
a)
b)
2. Solve the following equations:
9
 8
x2
x
7

 1
b) 2
x 9 x3
2
c) x  x  7 x  16
a) x 
3. Solve the inequality and write the solution using interval notation:
2x  5  7
a)
7  6x  3
b)
c) x  x  12  0
2
d) 2 x  5 x  12  0
4. Solve the following rational inequality:
2
5
1
x3
x 1
0
b)
x5
a)
5. Graph the following function:
a) f (x)  | x  3 | 2 ;
b) g ( x)  x  2, x  0 and
g ( x)  x 2  3, x  0 (piecewise defined function)
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Page 2
MAC 1105 Final Review - M. RAHMAN
c) h(x) 
Page 3
 ( x  3) 2  2
d) k ( x)  x  2  3
6. Find an equation of the line that passes through the point (3, -1) and is
perpendicular to the line 2 x  3 y  5 . Write the equation in slope-intercept form.
7. Solve the following system using substitution/elimination method method:
4 x  3 y  1
2x  5 y  3
8. Decide whether or not the following equation has a circle as its graph. If it does, give the center and the radius:
x 2  8 x  y 2  6 y  16  0
9. If g ( x)  5 x  2, calculate g (2), g ( 2  h), and
g ( x  h)  g ( x )
.
h
10. Evaluate ( f  g ) and ( g  f ) for the following functions
f (x)  x 2  5 ;
g ( x) 
x2
11. Find domain and range of f , g , ( f  g ), and ( g  f ) for the following function
f (x)  x  5 ;
7
g ( x)  2
x 4
9. Find the vertex, domain, and range of the following quadratic functions
a) h(x) 
 ( x  3) 2  2
b) h( x)  5 x  10 x  3
-----------------------------------------------------Part II: Circle the Correct Choice
2
1. Given
f ( x)  2 x 2  6 x  5 , calculate f (1) .
(a) 12
(b) -3
(c) 9
(d) 1
3x  1
, find g ( a  1) and simplify.
2 x
3a  2
3a  2
3a  2
(a)
(b)
(c)
3 a
3 a
3 a
(e) none of these
2. Given g ( x ) 
3. Given the following graph of f ( x ) 
(d)
3a  2
2a
(e) none of these
( x 2  4) , determine the open interval(s)
in which the function is increasing or decreasing.
(a) increasing on (4, ), decreasing on (-, -2)
(b) increasing on (0, 4), decreasing on (0, 4)
(c) increasing on (2, ), decreasing on (-, 0)
(d) increasing on (0, ), decreasing on (-, -2)
(e) none of these
4. Determine whether the function f ( x )  3 x is even, odd, or neither.
(a) even (b) odd
(c) neither
(d) both even and odd
5. Describe the transformation that you apply to the graph of
graph of g ( x)   | x  2 | .
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(e) none of these
f ( x )  x to obtain the
MAC 1105 Final Review - M. RAHMAN
(a)
(b)
(c)
(d)
(e)
Page 4
shift 2 unit to the left, then reflect in the x axis
shift 2 unit to the right, then reflect in the y axis
shift 2 unit to the right, then reflect in the x axis
shift 2 unit to the left, then reflect in the y axis
none of these
6. The following graph of g(x) is the transformation of the graph of
an equation for g(x) .
f ( x )  x 2 . Find
(a)
g ( x )  ( x  3) 2  1
(b)
g ( x )  ( x  1) 2  3
(c)
g ( x )  ( x  3) 2  1
(d) g ( x )  ( x  1)
(e) none of these
2
3
New materials(sample problems):
Rational function + Inverse function + Chap 4(Exponential and Logarithmic function):
1. Graph the following rational functions by showing its asymptotes, intercepts, and few points:
2x  1
a. f ( x) 
.
x3
x 1
b. g ( x) 
.
x4
2. For the function f ( x)  2 x  4 , check whether the function is one-to-one or not. If one-toone write an equation for the inverse. Give the domain and range of f(x) and f 1 ( x)
3. a. Graph the function f ( x)  2 x  1 .
b. Find domain, range, and horizontal asymptote.
c. Solve for x: 2 3 y  8
d. Solve for x: 4 x 2  2 3 x 3
4. a. Graph the function f ( x)  log 2 x .
b. Find domain, range, and vertical asymptote.
c. Write the following expression as a sum and/or difference of logarithms. Express
x3 x  1
power as factor: log
( x  2) 2
d. Solve for x: -2 log 4 x  log 4 9
e. Solve for x: log 4 ( x  6)  log 4 ( x  2)  log 4 x
f.
+ class notes + home work + quiz
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