Precalculus
Semester Exam Review
Identify intervals on which the function is increasing, decreasing, or constant.
1) g(x) = 1.25(x + 6)2
Perform the requested operation or operations. Find the domain.
2) f(x) = 4x + 1; g(x) = 4x - 5
Find f/g.
Find the domain of the given function (algebraically).
x
3) f(x) =
x2 + 3x
Perform the requested operation or operations. Find the domain of the composition.
4) f(x) = x2 + 1; g(x) = x - 2
Find f(g(x)).
Graph the function on your calculator to determine the domain and range from the graph.
5) k(x) = ex + 5
Solve the equation graphically.
6) 5x2 = x
Find the asymptote(s) of the given function.
x+2
7) g(x) =
horizontal asymptotes(s)
x2 - 8
Perform the requested operation or operations. Find the domain.
8) f(x) = x + 2; g(x) = cos x
Find f - g.
Perform the requested operation or operations. Find the domain of the composition.
1
9) f(x) = f(x) =
; g(x) = x
x-7
Find g(f(x)).
Solve the equation algebraically.
10) x - 6x - 9 = 0
Find the domain of the given function (algebraically).
11) f(x) = x2 + 7
Find the asymptote(s) of the given function.
x-9
12) f(x) =
vertical asymptotes(s) algebraically
x2 - 25
1
Solve the problem.
13) Determine graphically the local maximum and local minimum of f(x) = -4x2/3 + 1.
Determine whether the equation defines y as a function of x.
14) x = y2 - 5
Find the range of the function.
15) f(x) = 7 - x2
Describe how the graph of y=x2 can be transformed to the graph of the given equation.
16) y = (x - 17)2 - 6
Sketch the graph of y1 as a solid line or curve. Then sketch the graph of y2 as a dashed line or curve.
17) y1 = ∣x∣; y2 = -∣x+4∣
y
10
5
-10
-5
5
10
x
-5
-10
Find the inverse of the function.
18) f(x) = 5x + 1
Determine if the function is a power function. If it is, then state the power and constant of variation.
19) f x = 3x1/5
20) f x = 2 · 5 x
Write the statement as a power function equation. Use k as the constant of variation.
21) John kept track of the time it took him to drive to college from his home and the speed at which he drove. He
found that the time t varies inversely as the speed r.
22) The surface area of a sphere S varies directly as the square of its radius r.
2
Describe how to transform the graph of an appropriate monomial function f(x) = xn into the graph of the given
polynomial function. Then sketch the transformed graph.
23) g x = 3 x + 2 4 + 3
10
-10
10
-10
Describe the end behavior of the polynomial function by finding lim f x and lim f x .
x→∞
x→-∞
24) f x = -3x4 - 5x2 - 7
25) f x = x3 + 2x2 + 5x - 7
Find the zeros of the function algebraically.
26) f x = 3x2 + 2x - 1
Find the zeros of the function.
27) f x = 7x3 - 49x2 + 70x
Find the zeros of the polynomial function and state the multiplicity of each.
28) f(x) = (x + 5)2 (x - 1)
29) f(x) = 5(x + 7)2 (x - 7)3
Divide f(x) by d(x), and write a summary statement in the form indicated.
30) f x = x4 - 2x3 - 7x2 + 16x + 24; d x = x2 - 2x - 3
(Write answer in polynomial form)
Divide using synthetic division, and write a summary statement in fraction form.
2x3 + 3x2 + 4x - 10
31)
x+1
Find the remainder when f(x) is divided by (x - k)
32) f(x) = 4x2 - 2x - 3; k = 5
3
Use the Factor Theorem to determine whether the first polynomial is a factor of the second polynomial.
33) x + 2; 7x4 + 15x3 - 2x2 + x + 4
Use synthetic division to determine whether the number k is an upper or lower bound (as specified) for the real zeros of
the function f.
34) k = 2; f x = 3x3 + 2x2 + 5x + 6; Upper bound?
35) k = -4; f x = x3 - 3x2 + 5x + 4; Lower bound?
Find all of the real zeros of the function. Give exact values whenever possible. Identify each zero as rational or irrational.
36) f x = x3 - 5x2 - 13x + 65
Write the polynomial in standard form and identify the zeros of the function.
37) f(x) = (x - 5i)(x +5i)
38) f(x) = (x + 2)(x + 2)(x + 3i)(x - 3i)
Write a polynomial function of minimum degree with real coefficients whose zeros include those listed. Write the
polynomial in standard form.
39) 4, -2, and -1 + 2i
State how many complex and real zeros the function has.
40) f x = x3 - 18x2 + 89x - 72
Write a linear factorization of the function.
41) f(x) = x3 + 2x2 + 2x - 5
Write the function as a product of linear and irreducible quadratic factors, all with real coefficients.
42) f x = x3 - 6x2 - 3x - 28
State the domain of the rational function.
4
43) f(x) =
6-x
44) f(x) =
x-9
x2 + 9
Use limits to describe the behavior of the rational function near the indicated asymptote.
3
45) f(x) =
x-4
Describe the behavior of the function near its vertical asymptote.
4
Describe how the graph of the given function can be obtained by transforming the graph of the reciprocal function fx =
1/x.
3x - 1
46) f x =
x+3
For the given function, find all asymptotes of the type indicated (if there are any)
(x - 5)(x + 9)
47) f(x) =
, vertical
x2 - 9
48) f(x) =
49) f(x) =
x2 + 7x - 5
, slant
x-4
x+9
x2 + 7x + 7
, horizontal
Solve the equation.
6
50) x + 5 =
x
51)
8x
4
32
=
x-8 x
x2 - 8x
52)
2x
5
8
+
=
x+2 x-5
2
x -3x - 10
53)
x
2x - 3
-2x
=
+
2x + 2 4x + 4
x+1
Write an equation for the linear function f satisfying the given conditions.
54) f(3) = 5 and f(6) = 6
Write an equation for the quadratic function whose graph contains the given vertex and point.
55) Vertex (3, 1), point (0, 28)
Describe how to obtain the graph of the given monomial function from the graph of g(x) = xn with the same power n.
56) f x = -2.4x6
Determine if the function is a power function. If it is, then state the power and constant of variation.
1
57) f x = - x4
4
Write the quadratic function in vertex form.
58) y = x2 + 6x + 2
5
Describe the end behavior of the polynomial function by finding lim f x and lim f x .
x→-∞
x→∞
59) f x = 5x4 + 3x2 + 5
Find the zeros of the function algebraically.
60) f x = x2 + 6x + 5
Describe how to transform the graph of an appropriate monomial function f(x) = xn into the graph of the given
polynomial function. Then sketch the transformed graph.
61) g x = - x + 6 3
Find the vertex and axis of symmetry of the graph of the function.
62) f(x) = (x + 1)2 - 5
Decide if the function is an exponential function. If it is, state the initial value and the base.
63) y = 7 x
Compute the exact value of the function for the given x-value without using a calculator.
64) f(x) = 6 x for x = 2
65) f(x) = 3 1-x for x = 4
6
Graph the function. Describe its position relative to the graph of the indicated basic function.
66) f(x) = -2 x+3 ; relative to f(x) = 2 x
y
10
5
-10
-5
5
10
x
-5
-10
Decide whether the function is an exponential growth or exponential decay function and find the constant percentage
rate of growth or decay.
67) f(x) = 20,026 · 0.878x
Find the exponential function that satisfies the given conditions.
68) Initial value = 32, increasing at a rate of 15% per year
69) Initial population = 1184, doubling every 9 hours
Solve the problem.
70) The population of a small country increases according to the function B = 2,200,000e0.02t, where t is measured
in years. How many people will the country have after 5 years?
71) A bacterial culture has an initial population of 10,000. If its population declines to 4000 in 6 hours, what will it
be at the end of 8 hours?
Evaluate the logarithm.
1
72) log6 ( )
6
Simplify the expression.
73) log10 104
Solve the equation by changing it to exponential form.
74) log x = -4
4
7
Graph the function. Describe its position relative to the graph of the indicated basic function.
75) f(x) = ln(x - 4); relative to f(x) = ln x
y
10
5
-10
-5
5
10
x
-5
-10
Rewrite the expression as a sum or difference or multiple of logarithms.
15 x
76) log14
y
Use the product, quotient, and power rules of logarithms to rewrite the expression as a single logarithm. Assume that all
variables represent positive real numbers.
77) 7ln (xy) - 4ln (yz)
Rewrite the expression as a sum or difference or multiple of logarithms.
78) ln x2 y3
Use the change of base rule to find the logarithm to four decimal places.
79) log 95.10
9
Describe how to transform the graph of g(x) = ln x into the graph of the given function. Graph the function.
80) f(x) = log3x
6
y
4
2
-6
-4
-2
2
4
6 x
-2
-4
-6
Write the expression using only the indicated logarithms.
81) log9 x using natural logarithms
8
Solve the problem.
82) A cake is removed from an oven at 325 °F and cools to 150 °F after 25 minutes in a room 68 °F. How long will it
take the cake to cool to 120 °F?
83) Suppose you contribute $60 per month into a fund that earns 7.18% annual interest. What is the value of your
investment after 24 years?
84) How long must $5800 be in a bank at 8% compounded annually to become $10,735.40? (Round to the nearest
year.)
85) Matthew obtains a 30-year $88,000 house loan with an APR of 7.99%. What is his monthly payment?
86) How long will it take for $9600 to grow to $42,100 at an interest rate of 8.9% if the interest is compounded
continuously? Round the number of years to the nearest hundredth.
87) Find the interest rate necessary for a present value of $9510 to grow to a future value of $11,142.48 if interest is
compounded quarterly for two years.
Solve the equation.
88) log (x + 9) = 1 - log x
89) log 5 x = log 3 + log (x + 2 )
Solve the problem.
90) Find the present value of a loan with an annual interest rate of 6.6% and periodic payments of $1290.65 for a
term of 29 years, with payments made and interest charged 12 times per year.
91) Find the periodic payment of a loan with present value $16,000 and an annual interest rate 6% for a term of 5
years, with payments made and interest charged 12 times per year.
92) MAKE SURE YOU STUDY THE VOCABULARY WORDS ON THE LEARNING TARGET SHEET!
9
Answer Key
Testname: SEMESTER EXAM REVIEW SHEET
1) Increasing: (-6, ∞); decreasing: (-∞, -6)
4x + 1
5
2) (f/g)(x) =
; domain {x|x ≠ }
4x - 5
4
3) (-∞, -3) ∪ (-3, 0) ∪ (0, ∞)
4) f(g(x)) = x - 1
5) Domain: (-∞, ∞); range: (5, ∞)
1 1
6) 0, - ,
5 5
7) y = 0
8) x + 2 - cos x; domain: [-2, ∞)
1
9) g(f(x)) =
x-7
10) 3
11) (-∞, ∞)
12) x = 5, x = -5
13) Local maximum: 1; no local minimum
14) No
15) (-∞, 7]
16) Shift the graph of y = x2 right 17 units and then down 6 units.
17)
y
10
5
-10
-5
5
10
x
-5
-10
18) f-1(x) =
19) Power is
x2 - 1
for x ≥ 0
5
1
; constant of variation is 3
5
20) Not a power function
k
21) t =
r
22) S = kr2
10
Answer Key
Testname: SEMESTER EXAM REVIEW SHEET
23) Obtain the graph of g x = 3 x + 2 4 + 3 by shifting the graph of g x = x4 two units to the left, vertically stretching by a
factor of 3 and vertically shifting three units up.
10
-10
10
-10
24) -∞, -∞
25) ∞, -∞
1
26) and -1
3
27) 0, 5, and 2
28) -5, multiplicity 2; 1, multiplicity 1
29) -7, multiplicity 2; 7, multiplicity 3
30) f x = x2 - 2x - 3 x2 - 4 + 8x + 12
31) 2x2 + x + 3 +
-13
x+1
32) 87
33) No
34) Yes
35) Yes
36) 5 (rational), 13 (irrational), and - 13 (irrational)
37) f(x) = x2 + 25; zeros ± 5i
38) f(x) = x4 + 4x3 + 13x2 + 36x + 36; zeros -2 (mult. 2), ± 3i
39) f(x) = x4 - 7x2 - 26x - 40
40) 3 complex zeros; all 3 real
41) f(x) = (x - 1)(2x + 3 + 11i)(2x + 3 42) x - 7 x2 + x + 4
11i)
43) (-∞, 6) ∪ (6, ∞)
44) (-∞, ∞)
45) lim f(x) = -∞, lim f(x) = ∞
x→4 x→4 +
46) Shift the graph of the reciprocal function left 3 units, reflect across the x-axis, stretch vertically by a factor of 10, and
then shift 3 units up.
47) x = 3, x = -3
48) y = x + 11
49) y = 0
50) x = -6 or x = 1
1
51) x =
2
11
Answer Key
Testname: SEMESTER EXAM REVIEW SHEET
52) x =
1
or x = 2
2
53) x = 3
54) f(x) =
1
x+4
3
55) P(x) = 3x2 - 18x + 28
56) Vertically stretch by a factor of 2.4, and then reflect across x-axis
1
57) Power is 4; constant of variation is 4
58) y = (x + 3)2 - 7
59) ∞, ∞
60) -1 and -5
61) Obtain the graph of g x = - x + 6 3 by shifting the graph of g x = x3 six units to the left and rotating across the x-axis.
62) (-1, -5)
63) Exponential Function; base = 7; initial value = 1
64) 36
1
65)
27
66) Moved left 3 unit(s);
reflected across x-axis
y
10
5
-10
-5
5
10
x
-5
-10
67) Exponential decay function; -12.2%
68) f(t) = 32 · 1.15t
69) P(t) = 1184 · 2t/9
12
Answer Key
Testname: SEMESTER EXAM REVIEW SHEET
70) 2,431,376
71) 2947
72) -1
73) 4
1
74) x =
44
75) Moved right 4 units
y
10
5
-10
-5
5
10
x
-5
-10
76) log14 15 +
1
log14 x - log14 y
2
x7 y3
z4
77) ln
78) 2ln x + 3ln y
79) 2.0730
80) Vertically shrink by a factor of 1/ln 3
6
y
4
2
-6
-4
-2
2
4
6 x
-2
-4
-6
81)
ln x
ln 9
82) 35 min
83) $45,864.74
84) 8 yr
85) $645.10
86) 16.61
87) 8%
88) 1
13
Answer Key
Testname: SEMESTER EXAM REVIEW SHEET
89) 3
90) $199,871.29
91) $309.32
92)
14