MTH 112 Practice problems for Test 2.
Use the graph to determine the function's domain and
range.
1)
6
Find and simplify the difference quotient
f(x + h) - f(x)
, h≠ 0 for the given function.
h
y
4) f(x) =- x2 - 9x - 8
5
4
5) f(x) =
3
2
3
5x-2
1
-6 -5 -4 -3 -2 -1
-1
1
2
3
4
5
6) f(x) =
6 x
-2
6x+7
Identify the intervals where the function is increasing,
decreasing or contant.
7)
-3
-4
-5
-6
5
y
4
3
2)
2
6
y
1
5
-5
4
-4
-3
-2
-1
1
2
3
4
5 x
-1
3
-2
2
-3
1
-4
-6 -5 -4 -3 -2 -1
-1
1
2
3
4
5
6 x
-5
-2
-3
Use the graph of the given function to find any relative
maxima and relative minima.
8) f(x) = x3 - 12x + 2
-4
-5
-6
20
y
16
3)
12
10
y
8
8
4
6
-5
4
-6
-4
-2
-3
-2
-1
1
-4
2
-10 -8
-4
-8
2
4
6
8
-12
x
-2
-16
-4
-20
-6
-8
-10
1
2
3
4
5 x
9)
5
Use the vertex and intercepts to sketch the graph of the
quadratic function. Also find the axis of symmetry, the
domain, , the range, and the maximum or minimum.
17) f(x) = 1 - (x - 1)2
y
4
3
2
y
1
-5
-4
-3
-2
-1
10
1
2
3
4
5 x
-1
5
-2
-3
-10
-4
-5
-5
-5
Graph each of the following basic functions. Plot some
points
10) f(x) = x
-10
5
10
x
5
10
x
18) f(x) = x2 + 6x + 5
y
11) f(x) = x3
10
5
12) f(x) = x
-10
13) f(x) = x2
-5
-5
Find the domain of the function.
x
14) g(x) =
x2 - 16
15) f(x) =
-10
Graph the piecewise function .
19)
-3x - 5 if x < -1
f(x) =
2x + 4 if x ≥ -1
19 - x
Find the domain of the composite function f∘g.
4
16) f(x) = x + 1,
g(x) =
x+6
20)
x-3
f(x) = 3
x2 -1
2
if x > -2
-2≤ x<1
if x ≥ 1
36) f(x) = -5x4 + 5x3 + 5x + 1; between 1 and 2
Find the x-intercepts of the polynomial function. State
whether the graph crosses the x-axis, or touches the x-axis
and turns around, at each intercept.
21) f(x) = 7x2 - x3
Determine the maximum possible number of turning
points for the graph of the function.
37) f(x) = - x2 - 8x - 33
22) f(x) = x4 - 64x2
38) f(x) = 9x8 - 6x7 + -8x - 2
23) x4 + 4x3 - 96x2 = 0
39) f(x) = x3 ( x3 - 7)(4x + 5)
24) f(x) = -x2 (x + 6)(x2 - 1)
Determine end behavior, find the real zeros and their
multiplicity, the y-intercept, and using those graph the
polynomial function.
40) f(x) = x3 + 4x2 - x - 4
Use the Leading Coefficient Test to determine the end
behavior of the polynomial function.
25) f(x) = 3x4 - 5x3 - 4x2 - 3x - 5
y
26) f(x) = -2x4 - 5x3 + 5x2 + 5x + 5
27) f(x) = 3x3 + 2x2 - 4x + 1
28) f(x) = -3x3 - 3x2 - 2x + 1
x
Find the zeros of the polynomial function.
29) f(x) = x3 + x2 - 20x
30) f(x) = x3 + 2x2 - 9x - 18
Find the zeros for the polynomial function and give the
multiplicity for each zero. State whether the graph crosses
the x-axis or touches the x-axis and turns around, at each
zero.
31) f(x) = 4(x - 3)(x - 2)2
41) f(x) = 6x3 - 4x - x5
y
10
5
32) f(x) = -4 x +
33) f(x) =
5
(x - 5)3
2
-10
-5
5
-5
1 2 2
x (x - 3)(x + 1)
3
-10
34) f(x) = x3 + x2 - 20x
Divide using long division.
42) (-24x2 + 38x - 15) ÷ (-4x + 3)
Use the Intermediate Value Theorem to determine
whether the polynomial function has a real zero between
the given integers.
35) f(x) = 9x3 - 6x2 + 2x - 7; between 1 and 2
3
10
x
43)
8x3 + 18x2 + 13x + 5
-4x - 3
Solve the polynomial equation. In order to obtain the first
root, use synthetic division to test the possible rational
roots.
56) 2x3 - 11x2 + 17x - 6 = 0
Divide using synthetic division.
44) (x2 + 12x + 36) ÷ (x + 6)
45)
Find an nth degree polynomial function with real
coefficients satisfying the given conditions.
57) n = 3; - 4 and i are zeros; f(-3) = 60
-3x3 - 3x2 + 12x + 12
x+2
58) n = 3; 3 and i are zeros; f(2) = 30
x5 + x3 - 5
46)
x-2
Find the domain of the rational function.
6x2
59) f(x) =
(x - 1)(x - 5)
Use synthetic division and the Remainder Theorem to find
the indicated function value.
47) f(x) = x4 - 7x3 - 6x2 - 7x + 3; f(-3)
60) f(x) =
x+2
x2 + 9
48) f(x) = 7x4 + 3x3 + 6x2 - 4x + 73; f(2)
Use the graph of the rational function shown to complete
the statement.
61)
Solve the problem.
49) Use synthetic division to divide f(x) = x3 - 2x2
- 13x - 10 by x + 2. Use the result to find all
zeros of f.
10
y
8
6
Use synthetic division to show that the number given to
the right of the equation is a solution of the equation, then
solve the polynomial equation.
50) 2x3 + 3x2 - 5x - 6 = 0; -2
4
2
-10 -8 -6 -4 -2
-2
-4
-6
51) 5x3 - 24x2 + 25x + 6 = 0; 2
-8
-10
Use the Rational Zero Theorem to list all possible rational
zeros for the given function.
52) f(x) = x5 - 6x2 + 6x + 5
As x→-3 - , f(x)→ ?
53) f(x) = -2x3 + 4x2 - 3x + 8
Find a rational zero of the polynomial function and use it
to find all the zeros of the function.
54) f(x) = x3 + 2x2 - 9x - 18
55) f(x) = x3 + 3x2 + 4x - 8
4
2
4 6
8 10
x
62)
10
71) h(x) =
y
x3 - 8
x2 + 5x
8
6
Graph the rational function.
2x
72) f(x) =
x-1
4
2
-8
-6
-4
-2
2
4
6
8
10
x
y
-2
-4
5
-6
-8
-10
-10
As x→2 - , f(x)→ ?
-5
10 x
-5
Find the vertical asymptotes, if any, of the graph of the
rational function.
x+2
63) h(x) =
x(x + 5)
64) h(x) =
5
-10
x
x(x + 3)
73) f(x) =
x2
2
x - x - 56
y
65)
x - 25
x2 - 8x + 15
Find the horizontal asymptote, if any, of the graph of the
rational function.
15x
66) f(x) =
3x2 + 1
67) g(x) =
6x2
2x2 + 1
68) h(x) =
25x3
5x2 + 1
x
Find the slant asymptote, if any, of the graph of the
rational function.
x2 + 6x - 8
69) f(x) =
x-7
70) f(x) =
x2 + 9
x
5
Solve the polynomial inequality and graph the solution
set on a number line. Express the solution set in interval
notation.
74) (x - 2)(x + 5) > 0
-10-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
75) (x + 4)(x - 1) ≤ 0
-10-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
76) x2 - 5x - 14 < 0
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
77) x2 + 3x + 2 ≥ 0
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
78) (x + 6)(x + 3)(x - 3) < 0
-10-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
Solve the rational inequality and graph the solution set on
a real number line. Express the solution set in interval
notation.
x-2
79)
<0
x+1
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
80)
x-5
>0
x+3
-10-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
81)
16 - 4x
≤0
5x + 8
-10-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
6
Answer Key
Testname: MTH 112 TEST 2 PRACTICE PROBLEMS SPRING 2014
1) domain: (-∞, ∞)
range: [-4, ∞)
2) domain: [0, ∞)
range: [-2, ∞)
3) domain: (-∞, ∞)
range: [0, 6]
4) f(x+h) = -(x+h)2 -9(x+h)-8
y
x
=-(x2 +2xh+h 2 )-9(x+h)-8
= -x2 -2xh-h 2 -9x-9h-8
f(x+h)-f(x)= -x2 -2xh-h 2 -9x-9h-8-(- x2 - 9x - 8)
= -x2 -2xh-h 2 -9x-9h-8 +x2 +9x+8
= -2xh - h 2 - 9h
f(x+h)-f(x) -2xh - h 2 - 9h
=
h
h
11)
Standard cubic
function
y
h(-2x - h - 9)
=
= -2x - h - 9
h
5)
6)
-15
(5x+5h-2)(5x-2)
6
6x+6h+7 6x+7
x
7) increasing (-2, -1) or (3, ∞)
decreasing (1, 3)
constant (-1, 1)
8) minimum: (2, -14); maximum: (-2, 18)
9) relative minimum: (3, -1)
12)
y
Identity
function
y
x
x
10)
Square root
function
13)
quadratic function
14) (-∞, -4) ∪ (-4, 4) ∪ (4, ∞)
15) (-∞, 19]
16) (-∞, -6) ∪ (-6, ∞)
7
Standard
Answer Key
Testname: MTH 112 TEST 2 PRACTICE PROBLEMS SPRING 2014
17) vertex: (1,1)
axis of symmetry: x = 1
x-intercepts: (0,0) and (2,0)
y-intercept: (0,0)
domain: (-∞,∞)
range: (-∞, 1]
maximum: (1,1)
24) 0, touches the x-axis and turns around;
-6, crosses the x-axis;
-1, crosses the x-axis;
1, crosses the x-axis
25) rises to the left and rises to the right
26) falls to the left and falls to the right
27) falls to the left and rises to the right
28) rises to the left and falls to the right
29) x = 0, x = - 5, x = 4
30) x = -2, x = -3, x = 3
31) 3, multiplicity 1, crosses x-axis; 2, multiplicity 2,
touches x-axis and turns around
5
32) - , multiplicity 1, crosses x-axis; 5, multiplicity 3,
2
y
10
5
-10
-5
5
10
x
crosses x-axis
33) 0, multiplicity 2, touches x-axis and turns around;
-1, multiplicity 1, crosses x-axis;
3, multiplicity 1, crosses x-axis;
- 3, multiplicity 1, crosses x-axis
34) 0, multiplicity 1, crosses the x-axis
- 5, multiplicity 1, crosses the x-axis
4, multiplicity 1, crosses the x-axis
35) f(1) = -2 and f(2) = 45; yes
36) f(1) = 6 and f(2) = -29; yes
37) 1
38) 7
39) 6
-5
-10
18) vertex: (-3, -4)
axis of symmetry: x = -3
x-intercepts: (-5,0) and (-1,0)
y-intercept: (0,5)
domain: (-∞,∞)
range: [-4,∞)
minimum: (-3,-4)
y
y
10
100
80
5
60
40
-10
-5
5
10
x
20
-5
-10 -8
-6
-4
-2
-20
-40
-10
-60
19) Get help if you need.
20) Get help if you need it.
21) 0, touches the x-axis and turns around;
7, crosses the x-axis
22) 0, touches the x-axis and turns around;
8, crosses the x-axis;
-8, crosses the x-axis
23) 0, touches the x-axis and turns around;
-12, crosses the x-axis;
8, crosses the x-axis
-80
40)
8
-100
2
4
6
8
10 x
Answer Key
Testname: MTH 112 TEST 2 PRACTICE PROBLEMS SPRING 2014
41)
71) y = x - 5
72)
y
10
y
10
5
5
-10
-5
5
10
x
-10
-5
5
10
x
-5
-5
-10
-10
42) 6x - 5
43) -2x2 - 3x - 1 +
2
-4x - 3
73)
y
6
44) x + 6
45) -3x2 + 3x + 6
46) x4 + 2x3 + 5x2 + 10x + 20 +
5
4
35
x-2
3
2
47) 240
48) 225
49) {-2, -1, 5}
3
50)
, -1, -2
2
51) -
1
-16
-8
-1
8
16
x
-2
-3
-4
1
, 3, 2
5
-5
-6
52) ± 1, ± 5
1
53) ± , ± 1, ± 2, ± 4, ± 8
2
74) (-∞, -5) ∪ (2, ∞)
54) {-3, -2, 3}
55) {1, -2 + 2i, -2 - 2i}
1
56)
, 2, 3
2
-10-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
75) [-4, 1]
-10-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
57) f(x) = 6x3 + 24x2 + 6x + 24
58) f(x) = -6x3 + 18x2 - 6x + 18
76) (-2, 7)
59) {x|x≠ 1, x ≠ 5}
60) all real numbers
61) -∞
62) +∞
63) x = 0 and x = -5
64) x = -3
65) x = 5, x = 3
66) y = 0
67) y = 3
68) no horizontal asymptote
69) y = x + 13
70) y = x
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9
77) (-∞, -2] ∪ [-1, ∞)
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9
78) (-∞, -6) ∪ (-3, 3)
-10-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
9
Answer Key
Testname: MTH 112 TEST 2 PRACTICE PROBLEMS SPRING 2014
79) (-1, 2)
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9
80) (-∞, -3) or (5, ∞)
-10-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
81) -∞, -
8
or [4, ∞)
5
-10-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
10