John Jay College of Criminal Justice/CUNY
Department of Mathematics and Computer Science
MAT 105
FINAL EXAM REVIEW .
1.Evaluating and Simplifying Algebraic Expressions:
Evaluate the algebraic expression for the given value or values of the variable(s).
y - 7x
1)
;
x = -2 and y = 3
6x + xy
A) -
2)
17
18
B)
11
6
C) - 1
1)
D)
11
18
-b + b2 - 4ac
when a = 5, b = 14, and c = -3
2a
A)
1
5
B) -
1
5
2)
C) -3
D) 3
Simplify the algebraic expressions:
3) (12y + 9) + (11y2 - 6y + 9)
A) 11y2 + 18y - 18
3)
B) 29y6
4) (3a - 2b - 5c) - (9a - 6b - 7c)
A) -6a + 4b + 2c
B) 12a - 8b - 12c
5) (x - 11)(x 2 + 7x - 5)
A) x 3 - 4x 2 - 82x + 55
C) x 3 + 18x 2 + 72x - 55
6)
C) 11y2 - 6y + 18
4)
C) -6a + 4b - 12c
D) -6a - 8b + 2c
5)
B) x 3 + 18x 2 + 82x + 55
D) x 3 - 4x 2 - 72x - 55
-35x 2 + 28x + 21
7
A) -5x 2 + 4x + 3
C) -245x 2 + 196x + 147
D) 11y2 + 6y+ 18
6)
B) -35x 2 + 28x + 3
D) -5x 2 + 28x + 21
1
2. Exponential Expressions.
Simplify the exponential expressions:
7) (-6x 4 )(8x 7 )
A) -48x 28
8)
C) 48x 11
D) 48x 28
8)
B) 5x 5 y8
C) 5x 5 y8 z
D) x 5 y8 z
25x 13y6 0
5x 3 y3
A) x 10y3
10) (-5x 5 y-6 )(2x -1 y)
-10x 6
A)
y7
11)
B) -48x 11
20x 9 y11z 9
4x 4 y3 z 8
A) 5x 4 y7 z
9)
7)
9)
B) 5x 10y3
C) 1
D) 0
10)
B)
-3x 4
y5
C)
-10x 4
y5
D) -10x 4 y7
21x 13y13
7x 12y-10
A) 3xy3
11)
B) 3x 25y23
C) 3xy23
D) 21xy23
3. Radicals and Rational Exponents:
Evaluate the expression :
12) 144 + 25
A) 13
12)
B) 169
Add or subtract terms whenever possible.
13) 5 2 + 5 50
A) 10 2
B) 30 2
14)
2x + 6 8x - 2 32x
A) 5 42x
C) 17
D) 119
13)
C) -30 2
D) 20 2
14)
B) 4 42x
C) 5 2x
D) 4 2x
Rationalize the denominator.
3
15)
7- 2
A)
21 + 3 2
5
15)
B)
3
3
7
2
C)
21 - 3 2
47
D)
21 + 3 2
47
2
Simplify the radical expression.
3
16) x 8
3
A) x 2 x
16)
3
B) x x
3
C) x 2 x 2
3
D) x x 2
Evaluate the expressions :
17) 161/4
A) 8
18) 49-3/2
1
A)
343
17)
B) 16
C) 32
18)
B) -343
Simplify by reducing the index of the radical.
20
x 16
19)
4
5
A) x 4
B) x
20)
8
D) 2
C) 343
D) -
1
343
19)
C)
5
x4
D)
4
x
16x 4
A) 2 2x
20)
B)
1
4x
C)
4
2x
D) 2x
4. Factoring Polynomials:
Factor out the greatest common factor.
21) 21x 4 - 6x 3 + 15x 2
A) 3(7x 4 - 2x 3 + 5x 2 )
C) 3x(7x 3 - 2x 2 + 5x)
Factor by grouping.
22) x 3 + 9x - 3x 2 - 27
A) (x - 3)(x 2 + 9)
21)
B) x 2 (21x 2 - 6x + 15)
D) 3x 2 (7x 2 - 2x + 5)
22)
B) (x - 3)(x + 9)
C) (x - 3)(x 2 - 9)
D) (x + 3)(x 2 + 9)
Factor the trinomial, or state that the trinomial is prime.
23) x 2 - 12x + 27
A) (x + 9)(x - 3)
24) 6x 2 + 13x + 6
A) (6x + 2)(x + 3)
B) (x + 9)(x + 1)
23)
C) (x - 9)(x - 3)
24)
B) (3x - 2)(2x - 3)
C) (3x + 2)(2x + 3)
Factor the difference of two squares.
25) 49x 2 - 16y2
A) (7x + 4y)2
C) (7x - 4y)2
D) prime
D) prime
25)
B) (7x + 4y)(7x - 4y)
D) prime
3
5. Solving Linear and Absolute Value Equations and Inequalities.
Literal Equations.
Solve and check the linear equations.
26) (-5x + 4) - 5 = -4(x - 7)
A) {19}
B) {- 29}
27)
26)
C) {- 6}
D) {29}
2x x
= +5
5
3
A) {-75}
27)
B) {150}
Solve the linear inequality. Other than
on a number line.
28) 7x - 6 6x - 2
C) {75}
D) {-150}
, use interval notation to express the solution set and graph the solution set
28)
A) [4, )
B) (-8, )
C) (- , 4]
D) (- , 4)
4
29) -8x + 4
-2(3x + 1)
29)
A) [3, )
B) (- , 3)
C) (3, )
D) (- , 3]
Solve the compound inequality. Other than
solution set on a number line.
30) 17 5x - 3 22
, use interval notation to express the solution set and graph the
30)
A) (4, 5)
B) (-5, -4)
C) [4, 5]
D) [-5, -4]
5
Solve the absolute value inequality. Other than
solution set on a number line.
31) |x + 2| + 6 11
, use interval notation to express the solution set and graph the
31)
A) [-7, 11]
B) (-7, 3)
C) [-7, 3]
D) (- , -7]
[3, )
32) |7x - 9| - 3 > -6
32)
A) (- , )
B)
6 12
,
7 7
C)
6
,
7
D)
Solve the formula for the indicated letter.
9
33) F = C + 32 for C
5
A) C =
5
(F - 32)
9
B) C =
33)
9
(F - 32)
5
C) C =
5
F - 32
D) C =
F - 32
9
6
34) A =
1
bh, for b
2
A) b =
34)
Ah
2
B) b =
h
2A
C) b =
A
2h
D) b =
2A
h
6. Basics of Functions and their Graphs:
Determine whether the relation is a function.
35) {(-7, -1), (-7, 2), (-1, 8), (3, 3), (10, -7)}
A) Not a function
35)
B) Function
Evaluate the function at the given value of the independent variable and simplify.
36) f(x) = -3x - 8; f(-2)
A) 22
B) -2
C) 14
D) -11
37) f(x) = x + 11;
A) -3
C) 1.73
f(-2)
37)
B) 3
D) not a real number
Use the vertical line test to determine whether or not the graph is a graph in which y is a function of x.
38)
A) function
36)
38)
B) not a function
39)
39)
A) not a function
B) function
7
40)
40)
A) function
B) not a function
Use the graph to find the indicated function value.
41) y = f(x). Find f(-1)
A) -0.2
B) -4.2
41)
C) 4.2
D) 0.2
8
Use the graph to determine the function's domain and range.
42)
A) domain: (- , )
range: [-4, )
C) domain: [-1, )
range: [-4, )
42)
B) domain: (- , )
range: (- , )
D) domain: (- , -1) or (-1, )
range: (- , -4) or (-4, )
43)
43)
A) domain: [0, )
range: [0, )
C) domain: [0, )
range: [-1, )
B) domain: [0, )
range: (- , )
D) domain: (- , )
range: [-1, )
9
Identify the intervals where the function is changing as requested.
44) Increasing
A) (-3, 3)
B) (-3,
)
C) (-2,
44)
)
D) (-2, 2)
45) Constant
A) (-1, 1)
45)
B) (2, )
C) (-2, -1)
Evaluate the piecewise function at the given value of the independent variable.
46) f(x) = 3x + 1 if x < -1 ; f(2)
-2x - 5 if x -1
A) -8
B) -9
C) 1
D) (1, 2)
46)
D) -3
7. Linear Functions and Slope:
Find the slope of the line that goes through the given points.
47) (-2, -6), (-9, -17)
11
11
7
A)
B) C)
7
7
11
Use the given conditions to write an equation for the line in point-slope form.
48) Slope = 4, passing through (-3, 7)
A) x - 7 = 4(y + 3)
B) y = 4x + 19
C) y + 7 = 4(x - 3)
47)
D)
23
11
48)
D) y - 7 = 4(x + 3)
10
Use the given conditions to write an equation for the line in slope-intercept form.
2
49) Slope = , passing through (7, 3)
3
A) y =
2
x+7
3
B) y =
2
5
x3
3
50) Passing through (-8, -2) and (-5, -7)
5
46
A) y = - x 3
3
C) y = mx -
46
3
C) y = mx -
5
3
49)
D) y =
2
5
x+
3
3
50)
B) y + 2 = D) y =
Graph the line whose equation is given.
51) y = 2x - 2
5
(x + 8)
3
5
46
x3
3
51)
A)
B)
C)
D)
11
Determine the slope and the y-intercept of the graph of the equation.
52) 7x - 10y - 70 = 0
10
7
A) m =
B) m = 7; (0, 70)
C) m =
; (0, 10)
; (0, -7)
7
10
52)
D) m = -
Use the given conditions to write an equation for the line in the indicated form.
53) Passing through (2, 3) and parallel to the line whose equation is y = -2x + 3 ;
point-slope form
A) y - 2 = -2(x - 3)
B) y - 3 = -2(x - 2)
C) y = 2x
D) y - 3 = x - 2
54) Passing through (5, 3) and perpendicular to the line whose equation is y = 2x + 7;
point-slope form
1
1
A) y - 3 = (x + 5)
B) y - 3 = - (x - 5)
2
2
C) y - 5 =
1
(x - 3)
2
7
; (0, 7)
10
53)
54)
D) y = - 2x - 11
12
8. Transformations of Functions:
Begin by graphing the standard quadratic function f(x) = x 2 . Then use transformations of this graph to graph the
given function.
55) h(x) = (x - 7) 2 - 5
55)
A)
B)
C)
D)
13
Use the graph of the function f, plotted with a solid line, to sketch the graph of the given function g.
56) g(x) = -f(x - 1) + 2
56)
y = f(x)
A)
B)
C)
D)
9. Combinations of Functions; Composite Functions; Inverse Functions:
Given functions f and g, perform the indicated operations.
57) f(x) = 3 - 5x,
g(x) = -8x + 5
Find f + g.
A) -5x
B) 3x + 8
C) -8x + 3
For the given functions f and g , find the indicated composition.
58) f(x) = 3x + 9,
g(x) = 5x - 1
(f g)(x)
A) 15x + 8
B) 15x + 44
C) 15x + 6
57)
D) -13x + 8
58)
D) 15x + 12
14
59) f(x) = x2 + 2x + 2,
(f g)(-3)
A) 51
g(x) = x2 - 2x - 3
59)
B) 136
C) 170
Find the inverse of the one-to-one function.
60) f(x) = -4x + 5
x-5
A) f -1 (x) =
-4
C) f -1 (x) =
D) 17
60)
y-5
B) f -1 (x) =
-4
x+5
-4
-4x - 5
D) f -1 (x) =
-4
61) f(x) = x - 7
A) f -1 (x) = x 2 - 7
61)
B) f -1 (x) = x 2 + 7
C) f -1 (x) = x + 7
D) f -1 (x) =
1
x2 + 7
10. Distance and Midpoint Formula.
Find the distance between the pair of points.
62) (-1, 4) and (-5, 7)
A) 6
B) 5
62)
C) 10
Find the midpoint of the line segment whose end points are given.
63) (7, 3) and (4, 1)
3
A) ( , 1)
B) (11, 4)
C) (3, 2)
2
D) 25
63)
D) (
11
, 2)
2
11. Circles:
Write the standard form of the equation of the circle with the given center and radius.
64) (-4, 4); 3
A) (x + 4) 2 + (y - 4) 2 = 9
B) (x - 4) 2 + (y + 4) 2 = 9
C) (x + 4) 2 + (y - 4) 2 = 3
D) (x - 4) 2 + (y + 4) 2 = 3
Find the center and the radius of the circle.
65) (x - 5) 2 + (y + 7) 2 = 36
A) (-7, 5), r = 6
B) (7, -5), r = 36
64)
65)
C) (5, -7), r = 6
D) (-5, 7), r = 36
15
Graph the equation.
66) (x - 1) 2 + (y - 2) 2 = 49
66)
A)
B)
Domain = (-6, 8), Range = (-5, 9)
Domain = (-8, 6), Range = (-9, 5)
13. Complex Numbers:
Add or subtract as indicated and write the result in standard form.
67) -7 - (- 2 - 7i) - (- 2 + 5i)
A) 4 - 2i
B) -3 + 2i
C) -3 - 2i
Find the product and write the result in standard form.
68) (-3 - 7i)(2 + i)
A) -13 + 11i
B) -13 - 17i
67)
D) 4 + 2i
68)
C) 1 + 11i
D) 1 - 17i
Divide and express the result in standard form.
8
69)
4+i
A)
70)
32
8
i
17 17
B)
32
8
i
15 15
69)
C)
32
8
+
i
15 15
D)
32
8
+
i
17 17
6 - 6i
8 + 2i
A) 1 -
70)
1
i
4
B)
60 36
+
i
17 17
C)
3
1
- i
20 4
D)
9
15
i
17 17
16
Perform the indicated operations and write the result in standard form.
71) -16 + -81
A) -13
B) 13i
C) 36i
72)
71)
D) -13i
-2 - -24
2
A) 1 + i 6
72)
B) -1 - i 6
C) -1 + i 6
D) -1 - i 2
14. Quadratic Functions:
Solve the equation by factoring.
73) x 2 = x + 6
A) {-2, 3}
73)
B) {1, 6}
C) {-2, -3}
D) {2, 3}
Solve the equation by factoring.
74) x 2 + 2x - 120 = 0
A) {12, -10}
74)
B) {-12, 1}
C) {12, 10}
D) {-12, 10}
Solve the equation by the square root property.
75) 6x 2 = 54
A) {-3 6, 3 6}
76) (x - 3) 2 = 49
A) {52}
B) {-6, 6}
75)
C) {-3, 3}
D) {0}
76)
B) {-10, -4}
C) {-7, 7}
D) {-4, 10}
Solve the equation using the quadratic formula.
77) x 2 + 7x + 7 = 0
77)
A)
-7 - 21 -7 + 21
,
14
14
B)
-7 - 21 -7 + 21
,
2
2
C)
-7 - 77 -7 + 77
,
2
2
D)
7 - 21 7 + 21
,
2
2
78) 5x 2 - 3x + 3 = 0
3 ± i 51
A)
10
78)
B)
-3 ± 51
10
C)
-3 ± i 51
10
D)
3 ± 51
10
17
The graph of a quadratic function is given. Determine the function's equation.
79)
A) h(x) = (x - 2) 2 + 2
C) j(x) = (x - 2) 2 - 2
79)
B) g(x) = (x + 2) 2 - 2
D) f(x) = (x + 2) 2 + 2
80)
80)
A) f(x) = -x 2 - 2x - 1
C) j(x) = -x 2 + 1
B) g(x) = -x 2 + 2x + 1
D) h(x) = -x 2 - 1
Find the coordinates of the vertex for the parabola defined by the given quadratic function.
81) f(x) = (x - 4) 2 - 4
A) (0, -4)
82) y + 4 = (x - 2) 2
A) (2, - 4)
B) (4, 4)
C) (4, -4)
D) (-4, 0)
82)
B) (- 2, - 4)
C) (4, - 2)
D) (4, 2)
Find the axis of symmetry of the parabola defined by the given quadratic function.
83) f(x) = x2 + 7
A) x = -7
84) f(x) = (x + 4) 2 - 6
A) x = 6
81)
B) x = 7
C) x = 0
83)
D) y = 7
84)
B) x = -6
C) x = -4
D) x = 4
18
15. Polynomial and Rational Functions
Form a polynomial whose zeros and degree are given.
85) Zeros: -3, -2, 2; degree 3
A) f(x) = x 3 - 3x 2 + 4x - 12 for a = 1
C) f(x) = x 3 - 3x 2 - 4x + 12 for a = 1
85)
B) f(x) = x 3 + 3x 2 + 4x + 12 for a = 1
D) f(x) = x 3 + 3x 2 - 4x - 12 for a = 1
For the polynomial, list each real zero and its multiplicity. Determine whether the graph crosses or touches the
x-axis at each x -intercept.
86) f(x) = 5(x + 3)(x - 3) 3
86)
A) -3, multiplicity 1, crosses x-axis; 3, multiplicity 3, crosses x-axis
B) 3, multiplicity 1, touches x-axis; -3, multiplicity 3
C) 3, multiplicity 1, crosses x-axis; -3, multiplicity 3, crosses x-axis
D) -3, multiplicity 1, touches x-axis; 3, multiplicity 3
87) f(x) = 2(x 2 + 4)(x + 1) 2
A) -1, multiplicity 2, touches x-axis
B) -1, multiplicity 2, crosses x-axis
C) -4, multiplicity 1, crosses x-axis; -1, multiplicity 2, touches x-axis
D) -4, multiplicity 1, touches x-axis; -1, multiplicity 2, crosses x-axis
Find the x- and y-intercepts of f.
88) f(x) = (x + 4)(x - 2)(x + 2)
A) x-intercepts: -4, -2, 2; y-intercept: -16
C) x-intercepts: -4, -2, 2; y-intercept: 16
89) f(x) = 4x - x 3
A) x-intercepts: 0, -4; y-intercept: 0
C) x-intercepts: 0, 2, -2; y-intercept: 4
87)
88)
B) x-intercepts: -2, 2, 4; y-intercept: -16
D) x-intercepts: -2, 2, 4; y-intercept: 16
89)
B) x-intercepts: 0, 2, -2; y-intercept: 0
D) x-intercepts: 0, -4; y-intercept: 4
Find the domain of the rational function.
2x
90) g(x) =
x+2
A) {x|x
C) {x|x
91) f(x) =
B) all real numbers
D) {x|x 2}
x+7
x2 - 9
A) {x|x
C) {x|x
92) f(x) =
0}
-2}
90)
-3, x
-3, x
91)
3, x
3}
-7}
B) {x|x 0, x 9}
D) all real numbers
x+2
x 2 + 16x
A) {x|x
C) {x|x
-4, x 4}
0, x -16}
92)
B) all real numbers
D) {x|x -4, x 4, x
-2}
19
Find the vertical asymptotes of the rational function.
4x2
93) h(x) =
(x + 2)(x - 6)
A) x = 2, x = -6
C) x = -2, x = 6
94) g(x) =
93)
B) x = -2, x = 6, x = -4
D) x = -4
x+4
x2 + 4
94)
A) x = -2, x = 2
C) x = -2, x = -4
B) none
D) x = -2, x = 2, x = -4
List the potential rational zeros of the polynomial function. Do not find the zeros.
95) f(x) = 6x 4 + 2x 3 - 3x 2 + 2
95)
A) ±
1
1
1
2
, ± , ± , ± , ± 1, ± 2, ± 3
6
3
2
3
B) ±
1
1
1
, ± , ± , ± 1, ± 2
6
3
2
C) ±
1
1
1
2
, ± , ± , ± , ± 1, ± 2
6
3
2
3
D) ±
1
3
, ± , ± 1, ± 2, ± 3, ± 6
2
2
Use the Remainder Theorem to find the remainder when f(x) is divided by x - c.
96) f(x) = x 4 + 8x3 + 12x2 ; x + 1
A) R = 21
B) R = -21
C) R = -5
96)
D) R = 5
Form a polynomial f(x) with real coefficients having the given degree and zeros.
97) Degree 3: zeros: 1 + i and -5
A) f(x) = x3 + x2 - 8x + 10
B) f(x) = x3 -5x 2 - 8x - 12
C) f(x) = x3 + 3x 2 - 8x + 10
D) f(x) = x3 + 3x 2 + 10x - 8
97)
Use the given zero to find the remaining zeros of the function.
98) f(x) = x4 - 21x 2 - 100; zero: -2i
A) 2i, 5i, -5i
B) 2i, 10, -10
98)
C) 2i, 10i, -10i
D) 2i, 5, -5
16. Rational Expressions
Perform the indicated operations and simplify the result. Leave the answer in factored form.
4x - 4
8x 2
·
99)
x
5x - 5
A)
32x
5
B)
20x 2 + 40x + 20
8x 3
C)
32x 3 - 32x 2
5x 2 - 5x
D)
99)
5
32x
20
100)
x 2 - 10x + 24 x 2 - 8x + 15
·
x 2 - 4x + 3
x 2 - 14x + 48
100)
A)
(x + 4)(x + 5)
(x + 1)(x + 8)
B)
(x 2 - 10x + 24)(x 2 - 8x + 15)
(x 2 - 4x + 3)(x 2 - 14x + 48)
C)
(x - 4)(x - 5)
(x - 1)(x - 8)
D)
(x - 4)
(x - 8)
21