Practice problems for test 1
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Give the domain and range of the relation.
1) {(-7, -6), (-3, 2), (-5, 5), (-5, 8)}
1)
Determine whether the equation defines y as a function of x.
2) 2x + 2y = 6
2)
3) y = x3
3)
4) y = - x + 2
4)
Evaluate the function at the given value of the independent variable and simplify.
5) f(x) = -3x - 8; f(-2)
6) g(x) = 5x + 4;
7) f(x) =
x3 + 7
;
x2 + 5
5)
g(x + 1)
6)
f(-1)
7)
Graph the given functions on the same rectangular coordinate system. Describe how the graph of g is related to the graph
of f.
8) f(x) = x2 , g(x) = x2 + 4
8)
6
y
5
4
3
2
1
-6 -5 -4 -3 -2 -1
-1
1
2
3
4
5
6 x
-2
-3
-4
-5
-6
1
9) f(x) = 3x2 , g(x) = 3x2 - 4
6
9)
y
5
4
3
2
1
-6 -5 -4 -3 -2 -1
-1
1
2
3
4
5
6 x
-2
-3
-4
-5
-6
10) f(x) =
x, g(x) =
x+3
6
10)
y
5
4
3
2
1
-6 -5 -4 -3 -2 -1
-1
1
2
3
4
5
6 x
-2
-3
-4
-5
-6
Use the vertical line test to determine whether or not the graph is a graph in which y is a function of x.
11)
11)
y
x
2
12)
12)
y
x
13)
13)
y
x
Use the graph to find the indicated function value.
14) y = f(x). Find f(-2)
5
14)
y
4
3
2
1
-5
-4
-3
-2
-1
1
2
3
4
5 x
-1
-2
-3
-4
-5
3
Use the graph to determine the function's domain and range.
15)
6
15)
y
5
4
3
2
1
-6 -5 -4 -3 -2 -1
-1
1
2
3
4
5
6 x
-2
-3
-4
-5
-6
Identify the intervals where the function is changing as requested.
16) Increasing
5
16)
y
4
3
2
1
-5
-4
-3
-2
-1
1
2
3
4
5 x
-1
-2
-3
-4
-5
17) Increasing
17)
5
y
4
3
2
1
-10 -8
-6
-4
-2
2
4
6
8
10 x
-1
-2
-3
-4
-5
4
Use the graph of the given function to find any relative maxima and relative minima.
18) f(x) = x3 - 3x2 + 1
5
18)
y
4
3
2
1
-5
-4
-3
-2
-1
1
2
3
5 x
4
-1
-2
-3
-4
-5
Determine whether the given function is even, odd, or neither.
19) f(x) = x3 - 3x
19)
20) f(x) = 5x2 + x4
20)
21) f(x) = x3 + x2 - 4
21)
Evaluate the piecewise function at the given value of the independent variable.
if x > -2 ; f(-6)
22) f(x) = x + 3
-(x + 3) if x ≤ -2
22)
Graph the function.
x+5
23) f(x) = -4
-x + 5
23)
if -8 ≤ x < 2
if x = 2
if x > 2
y
10
5
-10
-5
5
10
x
-5
-10
Find and simplify the difference quotient
f(x + h) - f(x)
, h≠ 0 for the given function.
h
24) f(x) = x2 + 9x - 7
24)
5
Use the shape of the graph to name the function.
25)
25)
y
x
26)
26)
y
x
Begin by graphing the standard quadratic function f(x) = x 2 . Then use transformations of this graph to graph the given
function.
27) h(x) = (x - 5)2 + 3
27)
10
y
8
6
4
2
-10 -8 -6 -4 -2-2
2
4
6 8 10 x
-4
-6
-8
-10
6
28) g(x) = -
1
(x + 7)2 - 2
2
10
8
28)
y
6
4
2
-10 -8 -6 -4 -2-2
2
4
6 8 10 x
-4
-6
-8
-10
Begin by graphing the standard square root function f(x) =
function.
29) g(x) = - x + 1 + 2
10
x . Then use transformations of this graph to graph the given
29)
y
8
6
4
2
-10 -8 -6 -4 -2-2
2
4
6 8 10 x
-4
-6
-8
-10
Use the graph of the function f, plotted with a solid line, to sketch the graph of the given function g.
30) g(x) = -f(x - 2) - 2
30)
6
5
4
3
2
1
-6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
y
y = f(x)
1 2 3 4 5 6 x
Find the domain of the function.
3x
31) g(x) =
x2 - 81
31)
7
32) f(x) =
6-x
32)
Given functions f and g, perform the indicated operations.
33) f(x) = 7x - 9,
g(x) = 2x - 4
Find f - g.
33)
Given functions f and g, determine the domain of f + g.
2
34) f(x) = 5x + 10,
g(x) =
x-9
34)
For the given functions f and g , find the indicated composition.
35) f(x) = 4x2 + 2x + 5, g(x) = 2x - 4
35)
(g∘f)(x)
Find the domain of the composite function f∘g.
6
36) f(x) =
,
g(x) = x + 2
x+7
36)
Write the standard form of the equation of the circle with the given center and radius.
37) (8, -3); 9
38) (0, 0); 2 3
37)
38)
Find the center and the radius of the circle.
39) (x + 8)2 + (y - 1)2 = 4
39)
Graph the equation.
40) (x - 6)2 + (y - 3)2 = 9
40)
10
y
5
-10
-5
5
10 x
-5
-10
8
Graph the equation and state its domain and range. Use interval notation
41) x2 + y2 = 49
41)
y
10
5
-10
-5
10 x
5
-5
-10
Complete the square and write the equation in standard form. Then give the center and radius of the circle.
42) x2 - 2x + 1 + y2 - 6y + 9 = 25
42)
43) x2 + y2 + 10x + 8y = 8
43)
Use the vertex and intercepts to sketch the graph of the quadratic function.
44) f(x) = (x + 5)2 + 2
44)
y
10
5
-10
-5
5
10
x
-5
-10
45) f(x) = - 2x - 8 + x2
45)
y
10
5
-10
-5
5
10
x
-5
-10
9
Determine whether the given quadratic function has a minimum value or maximum value. Then find the coordinates of
the minimum or maximum point.
46) f(x) = -x2 - 2x + 1
46)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Use the Leading Coefficient Test to determine the end behavior of the polynomial function. Then use this end behavior
to match the function with its graph.
47) f(x) = 6x3 - 3x2 - 3x - 3
47)
A) rises to the left and rises to the right
B) falls to the left and falls to the right
y
y
5
5
4
4
3
3
2
2
1
1
-5 -4 -3 -2 -1
-1
1
2
3
4
5
x
-5 -4 -3 -2 -1
-1
1
2
3
4
5
x
-2
-2
-3
-3
-4
-4
-5
-5
C) falls to the left and rises to the right
D) rises to the left and falls to the right
y
y
5
5
4
4
3
3
2
2
1
1
-5 -4 -3 -2 -1
-1
1
2
3
4
5
x
-5 -4 -3 -2 -1
-1
-2
-2
-3
-3
-4
-4
-5
-5
10
1
2
3
4
5
x
48) f(x) = -6x3 - 2x2 + 3x + 2
A) rises to the left and rises to the right
48)
B) falls to the left and rises to the right
y
y
5
5
4
4
3
3
2
2
1
1
-5 -4 -3 -2 -1
-1
1
2
3
4
5
x
-5 -4 -3 -2 -1
-1
1
2
3
4
5
x
-2
-2
-3
-3
-4
-4
-5
-5
C) falls to the left and falls to the right
D) rises to the left and falls to the right
y
y
5
5
4
4
3
3
2
2
1
1
-5 -4 -3 -2 -1
-1
1
2
3
4
5
x
-5 -4 -3 -2 -1
-1
-2
-2
-3
-3
-4
-4
-5
-5
1
2
3
4
5
x
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Use the Leading Coefficient Test to determine the end behavior of the polynomial function.
49) f(x) = 5x3 - 5x3 - x5
49)
Find the zeros of the polynomial function.
50) f(x) = x3 + 2x2 - x - 2
50)
51) f(x) = 4(x - 4)(x + 2)4
51)
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.
52) f(x) = 3(x + 1)(x + 2)3
52)
Determine the maximum possible number of turning points for the graph of the function.
53) f(x) = 8x3 - 8x2 - 5x - 22
11
53)
Graph the polynomial function.
54) f(x) = x4 - 4x2
10
54)
y
8
6
4
2
-10 -8 -6 -4 -2
-2
2
4
6 8 10 x
-4
-6
-8
-10
Determine the constant of variation for the stated condition.
55) s varies directly as r, and s = 60 when r = 5.
55)
Solve.
56) The amount of time it takes a swimmer to swim a race is inversely proportional to the
average speed of the swimmer. A swimmer finishes a race in 50 seconds with an average
speed of 3 feet per second. Find the average speed of the swimmer if it takes 37.5 seconds
to finish the race.
56)
Write an equation that expresses the relationship. Use k for the constant of proportionality.
57) p varies jointly as q and r and inversely as the square root of a.
57)
Write an equation that expresses the relationship. Use k as the constant of variation.
58) r varies jointly as the square of s and the square of t.
58)
59) d varies jointly as b and the difference between p and h.
Solve the problem.
60) The voltage across a resistor is jointly proportional to the resistance of the resistor and the
current flowing through the resistor. If the voltage across a resistor is 12 volts for a resistor
whose resistance is 4 ohms and when the current flowing through the resistor is 3 amperes,
find the voltage across a resistor whose resistance is 7 ohms and when the current flowing
through the resistor is 5 amperes.
59)
60)
Determine whether the relation is a function.
61) {(-2, -2), (1, 7), (5, 7), (8, -5), (10, 3)}
61)
62) {(-4, 3), (-2, -9), (1, 7), (1, -8)}
62)
12
Answer Key
Testname: TEST 1 PRACTICE
1)
2)
3)
4)
5)
domain = {-5, -3, -7}; range = {5, 2, -6, 8}
y is a function of x
y is a function of x
y is a function of x
-2
6) 5x + 9
7) 1
8) g shifts the graph of f vertically up 4 units
6
y
5
4
3
2
1
-6 -5 -4 -3 -2 -1
-1
1
2
3
4
5
6 x
-2
-3
-4
-5
-6
9) g shifts the graph of f vertically down 4 units
6
y
5
4
3
2
1
-6 -5 -4 -3 -2 -1
-1
1
2
3
4
5
6 x
-2
-3
-4
-5
-6
13
Answer Key
Testname: TEST 1 PRACTICE
10) g shifts the graph of f 3 units to the left
6
y
5
4
3
2
1
-6 -5 -4 -3 -2 -1
-1
1
2
3
4
5
6 x
-2
-3
-4
-5
-6
11) function
12) not a function
13) not a function
14) 3.6
15) domain: [0, ∞)
range: [-1, ∞)
16) (-2, 2)
17) (3, ∞)
18) maximum: (0, 1); minimum: (2, -3)
19) Odd
20) Even
21) Neither
22) 3
23)
y
10
(2, 7)
5
-10
(-8, -3)
(2, 3)
-5
5
10
x
-5 (2, -4)
-10
24) 2x + h + 9
25) Standard cubic function
26) Absolute value function
14
Answer Key
Testname: TEST 1 PRACTICE
27)
10
y
8
6
4
2
-10 -8 -6 -4 -2
-2
-4
2
4
6 8 10 x
2
4
6 8 10 x
2
4
6 8 10 x
-6
-8
-10
28)
10
8
y
6
4
2
-10 -8 -6 -4 -2-2
-4
-6
-8
-10
29)
10
y
8
6
4
2
-10 -8 -6 -4 -2-2
-4
-6
-8
-10
30)
6
5
4
3
2
1
-6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
y
1 2 3 4 5 6 x
31) (-∞, -9) ∪ (-9, 9) ∪ (9, ∞)
32) (-∞, 6]
15
Answer Key
Testname: TEST 1 PRACTICE
33) 5x - 5
34) (-∞, 9) or (9, ∞)
35) 8x2 + 4x + 6
36) (-∞, -9) or (-9, ∞)
37) (x - 8)2 + (y + 3)2 = 81
38) x2 + y2 = 12
39) (-8, 1), r = 2
40)
10
y
5
-10
-5
5
10 x
-5
-10
Domain = (3, 9), Range = (0, 6)
41)
10
y
5
-10
-5
5
10 x
-5
-10
Domain = (-7, 7); Range = (-7, 7)
42) (x - 1)2 + (y - 3)2 = 25
(1, 3), r = 5
43) (x + 5)2 + (y + 4)2 = 49
(-5, -4), r = 7
16
Answer Key
Testname: TEST 1 PRACTICE
44)
y
10
5
-10
-5
5
10
x
5
10
x
-5
-10
45)
y
10
5
-10
-5
-5
-10
46) maximum; - 1, 2
47) C
48) D
49) rises to the left and falls to the right
50) x = -1, x = 1, x = - 2
51) x = 4, x = -2,
52) -1, multiplicity 1, crosses x-axis; -2, multiplicity 3, crosses x-axis
53) 2
54)
20
y
16
12
8
4
-8
-6
-4
-2
-4
2
4
6
8
x
-8
-12
-16
-20
55) k = 12
56) 4 feet per second
17
Answer Key
Testname: TEST 1 PRACTICE
57) p =
kqr
a
58) r = ks2t2
59) d = kb(p - h)
60) 35 volts
61) Function
62) Not a function
18