MTH 112 Practice Test # 2 (Sections 1.8, 1.9, 2.1-2.6)
Name___________________________________
Determine which two functions are inverses of each other. Show mathematical justification for your answer.
x + 8
x - 2
1) f(x) = g(x) = 2x + 8
h(x) = Inverse functions:__________________________
2
8
Find the inverse of the one-to-one function.
3
2) f(x) = x + 2
______________________________________
Graph f as a solid line and f-1 as a dashed line in the same rectangular coordinate space. Use interval notation to give the domain
and range of f and f-1 . Include a table of at least five values for f and f-1 below.
3) f(x) = x2 - 4, x ≥ 0
10
y
8
6
4
2
-10 -8
-6
-4
-2
2
4
6
8
x
-2
-4
-6
-8
-10
Write the standard form of the equation of the circle with the given center and radius.
4) (3, -9); 1
Find the center and the radius of the circle.
5) (x - 9)2 + (y + 8)2 = 25
Center: _______________
__________________________________
Radius: ________________
Complete the square and write the equation in standard form. Then give the center and radius of the circle.
6) x2 + y 2 + 2x + 12y = -28
Center:____________
Radius:____________
Divide and express the result in standard form.
2 + 3i
7)
5 + 3i
___________________________
1
For the given quadratic function, answer each of the following questions.
8) f(x) = -x2 + 4x - 6
a.) Without graphing, find the vertex. vertex:_____________
b.) Does the function have a minimum or maximum value? _________________ What is that value? ___________
c.) What is the functionʹs domain? domain:_____________
d.) What is the functionʹs range? range:______________
Find the coordinates of the vertex for the parabola defined by the given quadratic function.
9) f(x) = (x + 9)2 - 8 vertex: ___________________
Use the the information you find in parts a -d below to sketch the graph of the quadratic function.
10) f(x) = x2 - 2x - 3
a.) What is the vertex of f(x)? ___________
b.) Write the equation of the axis of symmetry. ______________
c.) Find the x-intercept(s), if any. _____________________
d.) Find the y-intercept. _____________
y
10
5
-10
-5
5
10
x
-5
-10
Find the zeros of the polynomial function. State the multiplicity of each zero and whether the graph crosses the x -axis, or touches
the x-axis and turns around, at each intercept.
11) f(x) = -x2 (x + 3)(x2 - 1)
zeros
multiplicity
behavior of f(x)
2
Use the Leading Coefficient Test to determine the end behavior of the polynomial function.
12) f(x) = 5x3 - 3x2 - 2x + 3
__________________________________
Sketch the polynomial function by using the information found in parts a -c.
13) f(x) = x3 + 8x2 - x - 8
a.) Determine the graphʹs end behavior.
b.) Find the x-intercepts , the multiplicity, and the behavior of the graph at each.
x-intercept
(zero)
multiplicity
behavior of f(x)
c.) Find the y-intercept.
y
x
Use the Intermediate Value Theorem to determine whether the polynomial function has a real zero between the given integers.
Show mathematical justification for your answer and explain your findings.
14) f(x) = 7x3 - 10x - 2; between -2 and -1
Divide using long division. Determine q(x) and r(x).
-15x3 + 32x2 - 4x - 4
15)
3x - 4
q(x):__________________
r(x):___________________
Divide using synthetic division. Determine q(x) and r(x).
16) (x2 + 8x + 9) ÷ (x + 5)
q(x):__________________
r(x):___________________
3
Use the Remainder Theorem to find the indicated function value.
17) f(x) = 3x4 + 10x3 + 3x2 - 3x + 65; f(-2)
_______________________
Use the Rational Zero Theorem to list all possible rational zeros for the given function.
18) f(x) = 7x4 - x2 + 2 ____________________________________
Find a rational zero of the polynomial function and use it to find all the zeros of the function.
19) f(x) = x3 - 5x2 + 7x + 13
zeros:__________________________
Find an nth degree polynomial function with real coefficients satisfying the given conditions.
20) n = 3; - 6 and i are zeros; f(-3) = 60
f(x) = _________________________________
Find the domain of the rational function. Write answer in interval notation.
x + 3
21) f(x) = x2 - 64
domain:_______________________
Use the graph of the rational function shown to complete the statement.
22)
10
y
8
6
4
2
-10 -8 -6 -4 -2
-2
2
4 6 8 10
x
-4
-6
-8
-10
As x→-2 - , f(x)→ ___________________
Find the vertical asymptotes, if any, of the graph of the rational function.
x + 2
23) h(x) = x2 - 4
Find the horizontal asymptote, if any, of the graph of the rational function.
15x2
24) g(x) = 5x2 + 1
Find the slant asymptote, if any, of the graph of the rational function.
x2 - 6x + 5
25) f(x) = x + 7
Vertical asymptote(s):___________________
Horizontal asymptote:__________________
Slant asymptote:___________________
Graph the rational function by using the information in parts a - e. Show work to the side and place answers underneath prompts.
4
26) f(x) = x - 2
2
x - x - 72
a.) Find the vertical asymptote(s), if any.
b.) Find the horizontal asymptote, if any.
c.) Find the x-intercept(s), if any.
d.) Find the y-intercept.
e.) Make a list of additional points to be used in your graph.
f.) Graph the function using the information above.
y
6
5
4
3
2
1
-12 -10 -8 -6 -4 -2
-1
2
4
6
8 10 12 x
-2
-3
-4
-5
-6
5
Answer Key
Testname: MTH 112 BTZ PRACTICE TEST #2 (SECT. 1.8, 1.9, 2.1‐2.6) FA07
1) None
2) f-1 (x) = x3 - 2
y
100
80
3)
10
60
y
40
8
20
6
4
-10 -8
-6
-4
-2
-20
2
-10 -8
-6
-4
-2
2
4
6
8
-40
2
4
6
-60
8
-2
-80
-4
-100
13)
14) f(-2) = -38 and f(-1) = 1; yes
12
15) -5x2 + 4x + 4 + 3x - 4
-6
-8
-10
f domain = (0, ∞); range = (-4, ∞)
f-1 domain = (0, ∞); range = (-4, ∞)
4) (x - 3)2 + (y + 9)2 = 1
5) (9, -8), r = 5
6) (x + 1)2 + (x + 6)2 = 9
16) x + 3 - 6
x + 5
17) 51
2
1
18) ± , ± , ± 1, ± 2
7
7
(-1, -6), r = 3
9
19
+ i
7)
34 34
19) {-1, 3 + 2i, 3 - 2i}
20) f(x) = 2x3 + 12x2 + 2x + 12
21) {x|x ≠ -8, x ≠ 8}
22) +∞
23) x = 2
24) y = 3
25) y = x - 13
26)
8) (2, -2)
9) (-9, -8)
10)
y
10
y
6
5
5
4
-10
-5
5
10
3
x
2
1
-5
-10
-12 -10 -8 -6 -4 -2
-1
-2
-3
11) 0, touches the x-axis and turns
around;
-3, crosses the x-axis;
-1, crosses the x-axis;
1, crosses the x-axis
12) falls to the left and rises to the right
-4
-5
-6
6
2
4
6
8 10 1