A
Name: _________________________________________
Period _____________
College Algebra
Cumulative Final Exam A
Multiple Choice: Choose the response that best answers each question. (2 points each)
NO CALCULATOR!
1) Rationalize the denominator:
a)
b)
c)
d)
7
8−√5
56+7√5
59
56−7√5
59
7
8
−
7
√5
56+7√5
−3
2) Find the average rate of change for 𝑓(𝑥) = −3𝑥 2 − 𝑥 from 5 to 6.
a) –194
b) 194
c) –34
d) 34
3) Which of the following polynomial functions might have the graph below?
a) 𝑓(𝑥) = 𝑥 2 (𝑥 − 2)(𝑥 − 1)
b) 𝑓(𝑥) = 𝑥 2 (𝑥 − 2)2 (𝑥 − 1)2
c) 𝑓(𝑥) = 𝑥(𝑥 − 2)(𝑥 − 1)2
d) 𝑓(𝑥) = 𝑥(𝑥 − 2)2 (𝑥 − 1)
A
3
2
4) State the domain of the composite function 𝑓(𝑔(𝑥)) given that 𝑓(𝑥) = 𝑥−1 and 𝑔(𝑥) = 𝑥.
a) (−∞, 0) ∪ (0, 1) ∪ (1, 2) ∪ (2, ∞)
b) (−∞, −2) ∪ (−2, 0) ∪ (0, ∞)
c) (−∞, 0) ∪ (0, 1) ∪ (1, ∞)
d) (−∞, 0) ∪ (0, 2) ∪ (2, ∞)
5) If (–3, 5) is the endpoint of a line segment and (1, 3) is its midpoint, find the other endpoint.
a) ( 1, 2)
b) (5, 1)
c) (–5, –1)
d) (–1, –2)
6) Given 𝑓(𝑥) = 2𝑥 2 + 3 and 𝑔(𝑥) = 4𝑥 − 2 find f – g.
a) 2𝑥 2 + 4𝑥 − 5
b) 2𝑥 2 − 4𝑥 + 1
c) 2𝑥 2 + 4𝑥 + 1
d) 2𝑥 2 − 4𝑥 + 5
7) Solve the following inequality:
𝑥 + 20
≤3
𝑥+6
a) (−∞, −6) ∪ (−1, ∞)
b) (−∞, −6) ∪ [−1, ∞)
c) [−6, −1]
d) (−∞, −1] ∪ (6, ∞)
A
3
8) Express the following as a single logarithm: 9 log √𝑥 + 3 log 𝑥 − 2 log 𝑥
2
a) −10 log 𝑥
b) −4 log 𝑥
c) 2 log 𝑥
d) 4 log 𝑥
9) Find a fourth degree polynomial with real coefficients that has zeros: 2, 3,
2i
a) 𝑥 4 − 5𝑥 3 + 8𝑥 2 − 10𝑥 + 12
b) 𝑥 4 − 5𝑥 2 + 6
c) 𝑥 4 − 5𝑥 3 + 6𝑥 2
d) 𝑥 4 − 5𝑥 3 + 8𝑥 2 − 10𝑥 − 12
10) Solve the following inequality. Express your answer in interval notation. Graph the solution set.
|7𝑥 − 6| − 4 > 0
10
a) (
7
, ∞)
2
10
7
7
b) (−∞, ] ∪ [
, ∞)
2 10
c) ( ,
7
7
)
2
10
7
7
d) (−∞, ) ∪ (
, ∞)
A
11) Find and simplify the different quotient of f,
𝑓(𝑥+ℎ)−𝑓(𝑥)
ℎ
, ℎ ≠ 0, for the function 𝑓(𝑥) = 2𝑥 2 − 3𝑥 + 1
a) 2𝑥 + 2ℎ − 3
b)
−𝑥 2 +2𝑥ℎ+ℎ2
ℎ
c) 4𝑥 + 2ℎ − 3
d) 4𝑥 + 2ℎ + 3
12) Find the inverse of the following function: 𝑓(𝑥) =
3𝑥 − 2
𝑓(𝑥) =
𝑥+5
a) 𝑓 −1 (𝑥) =
3𝑥−2
𝑥+5
3𝑥−2
𝑥+5
𝑥+5
b) 𝑓 −1 (𝑥) = 3𝑥−2
c) 𝑓 −1 (𝑥) =
3𝑥+2
d) 𝑓 −1 (𝑥) =
5𝑥+2
𝑥−5
3−𝑥
13) Write the standard form equation of the circle:
a) (𝑥 − 3)2 + (𝑦 − 2)2 = 9
b) (𝑥 + 2)2 + (𝑦 + 3)2 = 9
c) (𝑥 − 2)2 + (𝑦 − 3)2 = 9
d) (𝑥 + 3)2 + (𝑦 + 2)2 = 9
A
14) Determine the equation for the following piecewise function:
a) 𝑓(𝑥) = {
b) (𝑥) = {
c) (𝑥) = {
(𝑥 − 2)2 − 3, 𝑥 < −2
√𝑥 − 2, 𝑥 ≥ −2
(𝑥 + 2)2 − 3, 𝑥 < −2
√𝑥 − 2, 𝑥 ≥ −2
(𝑥 + 2)2 − 3, 𝑥 < −2
√𝑥 + 2, 𝑥 ≥ −2
d) (𝑥) = {
(𝑥 − 2)2 − 3, 𝑥 < −2
√𝑥 + 2, 𝑥 ≥ −2
15) The half-life of “silicone-32” is 710 years. If a sample presently contains 50 grams, which expression
determines how much “silicone-32” will remain after 1650 years?
a) 𝐴(𝑡) = 50𝑒 710𝑘 , where 𝑘 =
ln 0.5
1650
b) 𝐴(𝑡) = 50𝑒 1650𝑘 , where 𝑘 =
c) 𝐴(𝑡) = 50𝑒 710𝑘 , where 𝑘 =
ln 0.5
710
ln 0.5
d) 𝐴(𝑡) = 50𝑒 1650𝑘 , where 𝑘 =
710
ln 0.5
32
16) Find an equation for the line with the given properties. Express your answer in general form.
Parallel to the line 8𝑥 + 5𝑦 = 107; containing the point (9,6)
a) 5𝑥 + 8𝑦 = 6
b) 9𝑥 + 5𝑦 = 107
c) 8𝑥 − 5𝑦 = 102
d) 8𝑥 + 5𝑦 = 102
A
17) Determine the equation for the following graph:
a) 𝑓(𝑥) = 3(𝑥 − 2)3 + 1
b) 𝑓(𝑥) = 3(𝑥 + 2)3 + 1
1
c) 𝑓(𝑥) = 3 (𝑥 − 2)3 + 1
1
d) 𝑓(𝑥) = 3 (𝑥 + 2)3 + 1
18) Find the quadratic function in the form y  a( x  h)2  k that has a maximum point at (-1, 17) and passes
through (7, 1).
a) 𝑦 = 2(𝑥 + 1)2 + 17
1
b) 𝑦 = − 4 (𝑥 + 1)2 + 17
c) 𝑦 = (𝑥 − 7)2 + 1
d) 𝑦 = −(𝑥 − 1)2 + 17
19) Find the domain of the function:
a) {𝑥|𝑥 ≠ 2}
𝑥−2
𝑥 3 −64𝑥
b) all real numbers
c) {𝑥|𝑥 ≠ −8, 0, 8}
d) {𝑥|𝑥 ≠ 0}
20) The path of a diver is given by ℎ(𝑥) = −2𝑥 2 + 12𝑥 + 20, where h is the height (in feet) and x is the
horizontal distance (in feet) from the end of the diving board. What is the maximum height of the diver?
a) 3 ft
b) –3 ft
c) 74 ft
d) 38 ft