Math 119 Final Review [Focus on the highlighted ones.]
1.
a. Solve the following absolute value equation. If needed, write your answer as a
fraction reduced to lowest terms. ⃒9𝑦 + 7⃒ − 8 = 4
b. Solve the following absolute value equation. If needed, write your answer as a
fraction reduced to lowest terms. ⃒−5𝑥 + 7⃒ + 6 = 3
2.
a. Simplify the following expression, writing your answer with only positive exponents.
3
(𝑛3𝑚
−3 )
b. Simplify the following expression, writing your answer with only positive exponents.
3
(−2𝑥
)
𝑦−2
3.
2
a. Consider the following pair of points: (−5, −5) and (−3, 2)
Step 1. Determine the distance between the two points.
Step 2. Determine the midpoint of the line segment joining the pair of points.
b. Consider the following pair of points: (−2, −8) and (1, 1)
Step 1. Determine the distance between the two points.
Step 2. Determine the midpoint of the line segment joining the pair of points.
4.
a. Consider the following function. 𝑟 ( 𝑥 ) = −𝑥 2 + 8𝑥 − 15
Step 1. Find the vertex.
Step 2. Find the 𝑥-intercepts, if any. Express the intercept(s) as ordered pair(s).
b. Consider the following function. ℎ ( 𝑥 ) = −𝑥 2 + 8𝑥 − 12
Step 1. Find the vertex.
Step 2. Find the 𝑥-intercepts, if any. Express the intercept(s) as ordered pair(s).
5.
a. Consider the following piecewise-defined function.
3
𝑥 − 8 𝑖𝑓𝑥 < 2
𝑓(𝑥) = { 4
−2𝑥 + 4 𝑖𝑓𝑥 ≥ 2
Step 1. Evaluate this function at 𝑥 = −2.
Step 2. Evaluate this function at 𝑥 = 6.
b. Consider the following piecewise-defined function.
4𝑥 − 4 𝑖𝑓𝑥 < 1
𝑓(𝑥) = {
2𝑥 − 5 𝑖𝑓𝑥 ≥ 1
Step 1. Evaluate this function at 𝑥 = 1.
Step 2. Evaluate this function at 𝑥 = 0.
Step 3. Evaluate this function at 𝑥 = 2.
6.
a. Consider the following function. 𝑚(𝑥) = (𝑥 + 2)3
Step 1. Graph the original function by indicating how the more basic function has been
shifted, reflected, stretched, or compressed.
b. Consider the following function. 𝑤(𝑥) = (𝑥 + 4)3
Step 1. Graph the original function by indicating how the more basic function has been
shifted, reflected, stretched, or compressed.
7.
a. Consider the following function. 𝑛(𝑥) = −2 √𝑥 + 5 + 1
Step 1. Determine the more basic function that has been shifted, reflected, stretched, or
compressed.
Step 2. Graph.
b. Consider the following function. 𝑝(𝑥) = 3𝑥 2 − 2
Step 1. Determine the more basic function that has been shifted, reflected, stretched, or
compressed.
Step 2. Graph.
8.
a. Consider the following relation. 𝑇 = { ( −7, 1 ), ( −2, −10 ), ( −9, 3 ), ( 9, −7 ) }
Step 1. Find the inverse. Express your answer as a set of ordered pairs.
Step 2. Find the domain and range of the inverse. Express your answer as a set of
numbers.
b. Consider the following relation. 𝑇 = { ( −8, 4 ), ( 7, 3 ), ( −9, 10 ), ( 6, −9 ) }
Step 1. Find the inverse. Express your answer as a set of ordered pairs.
Step 2. Find the domain and range of the inverse. Express your answer as a set of
numbers.
9.
a. Find the inverse of the following function.
3
𝐴 ( 𝑥 ) = 3 √𝑥 + 2
b. Find the inverse of the following function.
𝑃 ( 𝑥 ) = 5𝑥 4 + 3
10.
Consider the graph of 𝑓 (𝑥).
What is the average rate of change
of 𝑓 (𝑥) from 𝑥1 = 4 to 𝑥2 = 7?
11.
Find any vertical and horizontal asymptotes, if any, for the following function,
2
𝑓(𝑥) = −𝑥 −𝑥− +6𝑥9− 5
12.
a. Use polynomial long division to rewrite the following fraction in the form 𝑞( 𝑥 ) +
𝑟( 𝑥 )
, where 𝑑( 𝑥 ) is the denominator of the original fraction, 𝑞( 𝑥 ) is the quotient,
𝑑( 𝑥 )
and 𝑟( 𝑥 ) is the remainder.
9𝑥3 − 3𝑥2 + 3𝑥 + 15
3𝑥 + 3
b. Use polynomial long division to rewrite the following fraction in the form 𝑞( 𝑥 ) +
𝑟( 𝑥 )
, where 𝑑( 𝑥 ) is the denominator of the original fraction, 𝑞( 𝑥 ) is the quotient,
𝑑( 𝑥 )
and 𝑟( 𝑥 ) is the remainder.
13.
14.
4𝑥3 − 4𝑥2 + 7𝑥 + 5
2𝑥 + 1
a. Solve the following elementary exponential equation.
1
75𝑥 − 1 = 49
b. Solve the following elementary exponential equation.
33𝑥 − 9 = 19
a. Solve the following logarithmic equation.
ln( 𝑥 + 5 ) − ln( 𝑥 ) = 4
b. Solve the following logarithmic equation.
ln( 𝑥 + 6 ) − ln( 𝑥 ) = 3
15.
a. Solve the following exponential equation.
3𝑥
𝑒 2𝑥 + 3 = 711
b. Solve the following exponential equation.
𝑒 𝑥 + 7 = 9𝑥
16.
Use the properties of logarithms to condense the following expression as much as
possible, writing the answer as a single term with a coefficient of 1. All exponents
should be positive.
3 log( 𝑥𝑦 3 ) + 2 log( 𝑥1 ) − 2 log( 𝑦 )
17.
a. Find the exact value of the expression without the use of a calculator.
tan( sin−1 ( −√3
))
2
b. Find the exact value of the expression without the use of a calculator.
csc( sec −1( −√2 ) )
18.
Determine all angles associated with the following point (𝑥, 𝑦) on the unit circle. Write
the exact radian answer in [0,2𝜋) and indicate remaining answers by using 𝑛 to
represent any integer.
1 √3
(𝑥, 𝑦) = (− , )
2 2
19.
Use trigonometric identities, algebraic methods, and inverse trigonometric functions, as
necessary, to solve the following trigonometric equation on the interval [0, 2𝜋)
2 cos( 𝑥 ) − √2 = 0
20.
Use the sum and difference identities to determine the exact value of the following
expression.
11𝜋
5𝜋
tan ( 12 ) − tan ( 12 )
11𝜋
5𝜋
1 + tan ( 12 )tan ( 12 )
21.
Use the sum and difference identities to determine the exact value of the following
expression.
sin( 2𝜋
− 𝜋4 )
3
22.
a. Use the following information to determine cos( 2𝑥 ).
cos( 𝑥 ) = 15 and tan( 𝑥 ) is positive
b. Use the following information to determine cos( 2𝑥 ).
2
cos( 𝑥 ) = −√6
and tan( 𝑥 ) is positive
23.
Solve for the remaining side of the triangle described below.
𝐶 = 60°, 𝑏 = 5, 𝑎 = 6
24.
Solve for the remaining angle and sides of the triangle described below.
𝐵 = 40°, 𝐴 = 10°, 𝑏 = 2
5
−19
1.
a. Correct Answer: 𝑦 = 9 or 9
b. Correct Answer: No Solution (ø)
2.
a. Correct Answer: 27𝑚3 𝑛9
b. Correct Answer: 4𝑥 6 𝑦 4
3.
a. Step 1. Correct Answer: √53
Step 2. Correct Answer: (−4, −3
)
2
b. Step 1. Correct Answer: 3√10
Step 2. Correct Answer: (−1
, −7)
2 2
4.
a. Step 1. Correct Answer: (4, 1)
Step 2. Correct Answer: Two: (5, 0), (3, 0)
b. Step 1. Correct Answer: (4, 4)
Step 2. Correct Answer: Two: (2, 0), (6, 0)
5.
a. Step 1. Correct Answer: −19
2
Step 2. Correct Answer: −8
b. Step 1. Correct Answer: −3
Step 2. Correct Answer: −4
Step 3. Correct Answer: −1
6.
a.
b.
7.
a. Step 1. Correct Answer: 𝑓(𝑥) = √𝑥,
left 5, vertical stretch 2, x-axis reflections, & up 1
Step 2. Correct Answer:
b. Step 1. Correct Answer: 𝑓(𝑥) = 𝑥 2 ,
vertical stretch 3 & down 2
Step 2. Correct Answer:
8.
a. Step 1. Correct Answer: 𝑇 −1 = { ( 1, −7 ), ( −10, −2 ), ( 3, −9 ), ( −7, 9 ) }
Step 2. Correct Answer: Domain: { 1, −10, 3, −7 }, Range: { −7, −2, −9, 9 }
b. Step 1. Correct Answer: 𝑇 −1 = { ( 4, −8 ), ( 3, 7 ), ( 10, −9 ), ( −9, 6 ) }
Step 2. Correct Answer: Domain: { 4, 3, 10, −9 }, Range: { −8, 7, −9, 6 }
9.
a. Correct Answer: 𝐴−1 = (𝑥 −3 2)
4
3
𝑥−3
b. Correct Answer: 𝑃 −1 = √ 5
10.
11.
Correct Answer: −2
3
Correct Answer: 𝑥 = 9
12.
a. Correct Answer: 3𝑥 2 − 4𝑥 + 5
b. Correct Answer: 2𝑥 2 − 3𝑥 + 5
13.
a. Correct Answer: −1
5
b. Correct Answer: 73
a. Correct Answer: 𝑥 = 0.0933
b. Correct Answer: 𝑥 = 0.3144
14.
15.
a. Correct Answer: 𝑥 ≈ −2.04
b. Correct Answer: 𝑥 ≈ 5.85
16.
Correct Answer: log(𝑥𝑦 7 )
17.
a. Correct Answer: −√3
b. Correct Answer: √2
2𝜋
18.
Correct Answer: 𝑠 = 3 + 2𝑛𝜋, 𝑛 ∈ ℤ
19.
Correct Answer: 𝜋4, 7𝜋
4
20.
Correct Answer: Does Not Exist
21.
Correct Answer: √6 +4 √2
22.
a. Correct Answer: −23
25
b. Correct Answer: 13
23.
Correct Answer: 𝑐 = √31
24.
Correct Answer: 𝐶 = 130°, 𝑎 = sin (40) , 𝑐 = sin (40)
2sin (10)
2sin (130)