MATH 2412
EXAM #1 REVIEW - 1.2, 1.3, 2.3, 2.4, 2.5
Name_______________________________________________________________________________________________
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) {(11, -3), (-2, -7), (-5, -6), (-5, 6)}
1)
Objective: (1.2) Find the Domain and Range of a Relation
Determine whether the relation is a function.
2) {(-9, 2), (-9, 5), (2, 7), (4, -1), (9, 4)}
2)
Objective: (1.2) Determine Whether a Relation is a Function
Determine whether the equation defines y as a function of x.
3) y = 5x + 3
3)
Objective: (1.2) Determine Whether an Equation Represents a Function
4) x2 + y2 = 36
4)
Objective: (1.2) Determine Whether an Equation Represents a Function
Evaluate the function at the given value of the independent variable and simplify.
5) g(x) = 4x + 4;
g(x + 1)
5)
Objective: (1.2) Evaluate a Function
6) f(x) = -2x + 1;
f(-3)
6)
Objective: (1.2) Evaluate a Function
Use the vertical line test to determine whether or not the graph is a graph in which y is a function of x.
7)
7)
Objective: (1.2) Use the Vertical Line Test to Identify Functions
1
8)
8)
Objective: (1.2) Use the Vertical Line Test to Identify Functions
Use the graph to find the indicated function value.
9) y = f(x). Find f(2).
9)
Objective: (1.2) Obtain Information About a Function from Its Graph
Use the graph to determine the function's domain and range.
10)
Objective: (1.2) Identify the Domain and Range of a Function from Its Graph
2
10)
Identify the intercepts.
11)
11)
Objective: (1.2) Identify Intercepts from a Function's Graph.
Identify the intervals where the function is changing as requested.
12) Increasing
12)
Objective: (1.3) Identify Intervals on Which a Function Increases, Decreases, or is Constant
13) Constant
13)
Objective: (1.3) Identify Intervals on Which a Function Increases, Decreases, or is Constant
3
14) Decreasing
14)
Objective: (1.3) Identify Intervals on Which a Function Increases, Decreases, or is Constant
The graph of a function f is given. Use the graph to answer the question.
15) Find the numbers, if any, at which f has a relative maximum. What are the relative
maxima?
15)
Objective: (1.3) Use Graphs to Locate Relative Maxima or Minima
Determine whether the given function is even, odd, or neither.
16) f(x) = x3 + x2 + 4
16)
Objective: (1.3) Identify Even or Odd Functions and Recognize Their Symmetries
17) f(x) = x3 - 2x
17)
Objective: (1.3) Identify Even or Odd Functions and Recognize Their Symmetries
4
Use possible symmetry to determine whether the graph is the graph of an even function, an odd function, or a function
that is neither even nor odd.
18)
18)
Objective: (1.3) Identify Even or Odd Functions and Recognize Their Symmetries
Evaluate the piecewise function at the given value of the independent variable.
19) f(x) = 3x + 3 if x < -4 ; f(-2)
4x + 2 if x -4
19)
Objective: (1.3) Understand and Use Piecewise Functions
Graph the function.
20) f(x) = -x + 3
2x - 3
if x < 2
if x 2
20)
Objective: (1.3) Understand and Use Piecewise Functions
Determine whether the given function is even, odd, or neither.
21) f(x) = 2x2 + x4
21)
Objective: (1.3) Identify Even or Odd Functions and Recognize Their Symmetries
Find and simplify the difference quotient
22) f(x) =
f(x + h) - f(x)
, h 0 for the given function.
h
1
5x
22)
Objective: (1.3) Find and Simplify a Function's Difference Quotient
5
23) f(x) = 6x + 7
23)
Objective: (1.3) Find and Simplify a Function's Difference Quotient
24) f(x) = x2 + 7x + 3
24)
Objective: (1.3) Find and Simplify a Function's Difference Quotient
Determine whether the function is a polynomial function.
2
25) f(x) = 9 x6
25)
Objective: (2.3) Identify Polynomial Functions
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.
26) f(x) = 2x2 + 2x + 2
26)
Objective: (2.3) Determine End Behavior
27) f(x) = -2x2 - 3x + 2
27)
Objective: (2.3) Determine End Behavior
Find the zeros of the polynomial function.
28) f(x) = x3 + x2 - 30x
28)
Objective: (2.3) Use Factoring to Find Zeros of Polynomial Functions
29) f(x) = 4(x + 2)(x - 1)3
29)
Objective: (2.3) Use Factoring to Find Zeros of Polynomial Functions
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.
30) f(x) = 5(x - 2)(x - 7)4
30)
Objective: (2.3) Identify Zeros and Their Multiplicities
31) f(x) =
1 2 2
x (x - 5)(x - 3)
2
31)
Objective: (2.3) Identify Zeros and Their Multiplicities
Determine the maximum possible number of turning points (rel. max. and rel. min) for the graph of the function.
32) f(x) = 9x8 + 8x7 - 8x - 11
32)
Objective: (2.3) Understand the Relationship Between Degree and Turning Points
33) f(x) = -x2 + 8x + 7
33)
Objective: (2.3) Understand the Relationship Between Degree and Turning Points
6
Graph the polynomial function.
34) f(x) = x4 - 9x2
34)
Objective: (2.3) Graph Polynomial Functions
35) f(x) = -2x3 (x - 3)2 (x - 1)
35)
Objective: (2.3) Graph Polynomial Functions
Divide using long division.
36) (x2 - 8x + 7) ÷ (x - 1)
36)
Objective: (2.4) Use Long Division to Divide Polynomials
Divide using synthetic division.
37) (x2 + 14x + 45) ÷ (x + 5)
37)
Objective: (2.4) Use Synthetic Division to Divide Polynomials
38)
-6x3 - 21x2 - 12x - 9
x+3
38)
Objective: (2.4) Use Synthetic Division to Divide Polynomials
Use synthetic division and the Remainder Theorem to find the indicated function value.
39) f(x) = x4 - 5x3 - 8x 2 - 4x + 9; f(4)
Objective: (2.4) Evaluate a Polynomial Using the Remainder Theorem
7
39)
Use the Rational Zero Theorem to list all possible rational zeros for the given function.
40) f(x) = x5 - 5x2 + 5x + 21
40)
Objective: (2.5) Use the Rational Zero Theorem to Find Possible Rational Zeros
41) f(x) = -2x3 + 3x 2 - 2x + 8
41)
Objective: (2.5) Use the Rational Zero Theorem to Find Possible Rational Zeros
Find a rational zero of the polynomial function and use it to find all the zeros of the function.
42) f(x) = x3 - 8x2 + 19x - 14
42)
Objective: (2.5) Find Zeros of a Polynomial Function
43) f(x) = x4 + 5x3 - 4x 2 - 16x - 8
43)
Objective: (2.5) Find Zeros of a Polynomial Function
Find an nth degree polynomial function with real coefficients satisfying the given conditions.
44) n = 3; 3 and 2 and -1 are zeros; f(1) = 8
Objective: (2.5) Use the Linear Factorization Theorem to Find Polynomials with Given Zeros
8
44)
Answer Key
Testname: REVIEW_EXAM_1
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
11)
12)
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19)
20)
domain = {11, -2, -5}; range = {-3, -7, -6, 6}
Not a function
y is a function of x
y is not a function of x
4x + 8
7
not a function
function
1.25
domain: [0, )
range: [-1, )
(1, 0), (0, -3)
(0, 5)
(-1, 1)
(- , 3)
f has a relative maximum at x = 0; the relative maximum is 2
Neither
Odd
Odd
-6
21) Even
-1
22)
5x (x + h)
23) 6
24) 2x + h + 7
25) No
9
Answer Key
Testname: REVIEW_EXAM_1
26) rises to the left and rises to the right
27) falls to the left and falls to the right
28)
29)
30)
31)
x = 0, x = - 6, x = 5
x = -2, x = 1,
2, multiplicity 1, crosses x-axis; 7, multiplicity 4, touches x-axis and turns around
0, multiplicity 2, touches x-axis and turns around;
3, multiplicity 1, crosses x-axis;
5, multiplicity 1, crosses x-axis;
- 5, multiplicity 1, crosses x-axis
32) 7
33) 1
34)
10
Answer Key
Testname: REVIEW_EXAM_1
35)
36) x - 7
37) x + 9
38) -6x2 - 3x - 3
39) -199
40) ± 1, ± 7, ± 3, ± 21
1
41) ± , ± 1, ± 2, ± 4, ± 8
2
42) {2, 3 + 2, 3 - 2}
43) {-1, 2, -3 + 5, -3 - 5}
44) f(x) = 2x3 - 8x 2 +2 x +12
11