MTH 175 – Precalculus
Final Exam Review
Monroe Community College
Functions and Inverse Functions
1) In each equation below, does y represent a function of x?
a) x = y2
c) y =
b) x2 + y = 4
x −3
Functions and Inverse Functions... pg 1
Graphing Techniques…………… pg 3
Trigonometry …………………… pg 7
Conic Sections …………………..pg 13
Applications…………………….. pg 14
d) y = ± x
2) Is each function below one-to-one? If so, find the inverse function and its domain.
f ( x ) = ( x − 4)
h(x) = x3 + 4
g(x) = 3x – 5
2
j ( x ) = 6x − 1
3) Which graphs below represent functions of x? one-to-one functions of x?
A
C
B
4) Find the domain and range of each function of x:
5
b) y = 2
a) y = 2 x − 5
x −9
c)
d)
5
°
5
4
3
2
1
4
3
2
1
-7 -6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
1 2 3 4 5 6 7
-7 -6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
1 2 3 4 5 6 7
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MTH 175 – Precalculus
5) If f(x) =
Final Exam Review
1
and g(x) = x2 + 3, find:
x
a) f(x + 1)
b) ( f D g )( x )
d) ( g D f )( x )
e)
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g ( x + h) − g( x )
h
f) ( g D f )( 3 )
c)
( f D g )( 5 )
For b, d, state the domain.
6) Sketch the graph of each:
⎧5
c) y = ⎨
⎩2 x
b) y = x 2 − 9
a) y = x 2 − 5
x>2
x≤2
7) The values of f (x) are given by the following table.
x
f (x)
0
3
1
7
2
12
3
9
4
5
5
0
a) Find ( f D f )( 0 )
b) Find f −1 ( 5 )
8) Sketch the graph of the inverse of each function (use the same set of axes):
B
A
5
4
3
2
1
-7 -6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
5
4
3
2
1
1 2 3 4 5 6 7
-7 -6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
1 2 3 4 5 6 7
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MTH 175 – Precalculus
Final Exam Review
Monroe Community College
9. Find the inverse function for f(x) given by:
x
f (x)
10.
−5 6
−1
3
0
2
6
2
−7
11
Solve for x to the nearest hundredth:
a) 4e −3 x = 17
b) ln ( 3 x + 11) − 4 = 0
Graphing Techniques
1) For f(x) = x5 + x3 – 6x find
a)
b)
c)
d)
2)
any x-intercepts
any symmetry
any intervals (approximately) on which the function is increasing
any relative maximum/minimum points (approximately)
Sketch each graph, and give the domain, range, any intercepts, and the
equations of any asymptotes. Indicate the intervals over which the function is
increasing or decreasing.
a) y = 3e x + 1
3)
b) y = 2− x
c) y = log ( x − 4 )
Give the equation of the function whose graph would result from taking the graph
of y = x , shifting it left 2 units, reflecting that graph across the x-axis, then
shifting the resulting graph up by 5 units.
4) For each function, determine:
i) the domain, range, intercepts (if any)
ii) vertical asymptotes (if any)
iii) horizontal or slant asymptotes (if any)
iv) symmetry (if any) and the, sketch each graph
v) tell any interval(s) on which the function is
increasing
vi) tell any interval(s) on which the function is
decreasing
x3
a) f ( x ) = 2
x −1
3x 2 + x − 5
b) g ( x ) =
x2 − 4
c) h ( x ) =
1− 5x
1 + 2x
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MTH 175 – Precalculus
Final Exam Review
5) Using the graph of f(x) =
a) g ( x ) =
1
+1
x
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1
, describe the transformation of f (x).
x
b) h ( x ) = −
1
x
c) j ( x ) =
1
x+2
6) Using the graphing calculator, approximate to one decimal place the value(s) of x
where f(x) = 3 when f(x) = x3 – 5x2.
7)
Use your calculator to approximate solution(s) for x,
a) If 0 ≤ x < π and tan x = 2.25
b) If 0 ≤ x <
π
2
and sin x = 0.218
c) If 0 ≤ x < 2π and csc x = −1.6
8) Determine the viewing rectangle dimensions that display approximately one
period of the function.
a) y= sin 2x
1
b) y = 3 cos x
5
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MTH 175 – Precalculus
Final Exam Review
Monroe Community College
9) Match each equation with its correct graph:
a) y = log x
b) y = log (x + 3)
c) y = log x + 3
(1)
d)
y = log x − 2
(2)
5
4
3
2
1
-6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
1
2
3
4
5
5
4
3
2
1
(3)
-6 -5 -4 -3 -2 -1
10)
5
4
3
2
1
-1
-2
-3
-4
-5
-6 -5 -4 -3 -2 -1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
5
4
3
2
1
(4)
1
-1
-2
-3
-4
-5
-6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
For each function, determine: the x and y intercepts; the coordinates of any
maximum or minimum points; and intervals on which the function is increasing or
decreasing.
a) f ( x ) =
ln x
x
b) f ( x ) = x ln x
c) f ( x ) = 2 x 2 e −0.5 x
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MTH 175 – Precalculus
11)
y = e1.25 x
y = −3 x + 5
y = x 3 − 2x 2 + x − 1
b)
y = −x 2 + 3x − 1
Solve the system graphically and algebraically.
a)
13)
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Find, to the nearest tenth, the coordinates of the point of intersection of:
a)
12)
Final Exam Review
x 2 + y 2 = 25
x + 3 y = 15
b)
y = x 3 − 2x 2 + x − 1
y = −x 2 + 3x − 1
What type(s) of symmetry do the graphs of the following functions have?
a) y = sin x
b) y = cos x
c)
y= x
d) y = sin x cos x
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MTH 175 – Precalculus
Final Exam Review
Monroe Community College
Trigonometry
1) Determine the exact value of sin θ, cos θ, and tan θ if ( −2, −2 ) is on the terminal side
of θ.
2) Evaluate exactly:
a) tan
⎛ 5π ⎞
b) cos ⎜ −
⎟
⎝ 3 ⎠
11π
6
Use your calculator to approximate:
11π
d) sin
e) cot 46°
2
⎛ 3π ⎞
c) sin ⎜ −
⎟
⎝ 4 ⎠
f) sec 1.35
3) Find exactly:
a) arcsin (1/2)
b) sin-1(-1/2)
c) cos-1 (-1/2)
d) tan (csc-12)
e) sin (arccos 1/2)
Approximate:
f) arctan 2.37
g) arcsin (-0.7)
h) arctan (sin 2)
i) arcsin (cos 20)
4) Find the reference angle for θ when θ = …
a)
7π
6
b)
5π
3
c)
11π
12
d) 305o e) 440º
5) Convert to radians, rounding to the nearest thousandth: 48.175º,
6) Convert to radians, exactly:
120º22’30’’
225º
7) Convert each to degrees, rounding to the nearest hundredth:
−7π
, 2
3
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MTH 175 – Precalculus
Final Exam Review
Monroe Community College
8) Convert each angle to (1) DºM’S’’ form, and (2) radians, to the nearest tenth:
a) 23.45º
9) If sin x = −
b) 52.7º
c) 112.853º
d) 158.233º
4
12
3π
3π
and cos y = − , and π < x <
, π <y<
, find the exact value
5
13
2
2
of:
a)
sin (x – y)
b) cos (x + y)
c)
cos 2x
d) tan
x
2
10) If sin x = 1/5 and tan x < 0, find the exact value of cos x.
11) If cos x = −
5
and sin x < 0, find the exact value of tan x.
13
12) Find the arc length s as shown on each circle, using the given radius and central
angle.
a) r = 1.8, θ = 1.92
s
θ
r=1.8
b) r = 3, θ =32°
s
32°
r=3
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MTH 175 – Precalculus
Final Exam Review
Monroe Community College
13) Find the arc length s, to the nearest tenth, as shown in the figure:
88°
s
r=3
14) Find the exact value of each. State any identities used.
⎛ π⎞
a) tan ⎜ − ⎟
⎝ 3⎠
c)
1
⎛ 2π ⎞
cot ⎜
⎟
⎝ 3 ⎠
⎛π ⎞
b) cos 2 ⎜ ⎟ + sin 2
⎝6⎠
⎛ −5π ⎞
d) csc ⎜
⎟
⎝ 6 ⎠
⎛π ⎞
⎜6⎟
⎝ ⎠
⎛ 3π ⎞
e) sec 2 ⎜
⎟ −1
⎝ 4 ⎠
15) Using the Law of Sines, solve for B, when A = 45º, b = 4, and a = 3.
16) Use the Law of Cosines to solve for b (to the nearest tenth) when a = 7.6, c = 9.2,
B = 46º.
17) For what values of x is the statement true:
a) arccos (cos x) = x
b) tan (arctan x) = x
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MTH 175 – Precalculus
Final Exam Review
Monroe Community College
18) For each function, sketch the graph, and give its domain and range.
a) f ( x ) =
1
π⎞
⎛
tan ⎜ 2 x + ⎟
2
8⎠
⎝
π⎞
⎛
b) g ( x ) = − cot ⎜ x + ⎟
2⎠
⎝
19) Determine the domain and range of each function:
a)
b)
c)
d)
e)
f)
f (x) = 2 cos x
f (x) = 3 sin x + 2
f (x) = ½ cos x – 4
f (x) = sec x
f (x) = tan x
f (x) = 2 cos x – 3
20) Sketch one cycle of the graph of each of the following, noting the amplitude, period
and phase shift (or horizontal shift).
Label the coordinates of the five key points of a cycle (x-intercepts, relative max.,
and relative min.)
3
⎛x
⎞
a) y = − sin ⎜ + π ⎟
2
⎝2
⎠
amplitude _____________________
period
_____________________
phase shift ____________________
3
⎛x
⎞
b) On the above graph sketch the graph of y = − csc ⎜ + π ⎟
2
⎝2
⎠
π⎞
⎛
c) y = 3 cos ⎜ 2 x + ⎟
2⎠
⎝
amplitude _____________________
period
_____________________
phase shift ____________________
π⎞
⎛
d) On the above graph (c), sketch the graph of y = 3 sec ⎜ 2 x + ⎟
2⎠
⎝
e) For each function above, what is the domain? The range?
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MTH 175 – Precalculus
Final Exam Review
Monroe Community College
21) Solve the following equations for x over the interval: [0,2π ]
a) cos 2x = sin x
b) (2 cos x + 1) (4 sin 2 x – 1) = 0 c) 2 sin 2 x – 3sinx – 2 = 0
22) Show that a) (tan x)(sin x) + cos x = sec x
b)
1 + sec( −θ )
= − csc θ
sin( − θ ) + tan( −θ )
23) Use the sum or difference identities to find the exact values of the sine, cosine, and
the tangent of the given angle.
a) 195º = 225º - 30º
24)
b)
11π
3π π
=
+
6
12
4
For each graph below, write a possible equation. From your equation, find the
amplitude (if any), period, phase shift (or horizontal shift) and vertical shift.
A.
B.
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MTH 175 – Precalculus
Final Exam Review
Monroe Community College
C.
25) Sketch the graphs of y = cos −1 (x), y = sin −1 (x), and y = tan −1 (x)
26) Solve the following equation (to the nearest hundredth):
cos x = arccos x
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MTH 175 – Precalculus
Final Exam Review
Monroe Community College
Conic Sections
1)
Identify each of the following conic sections and graph it.
For a circle, state:
The coordinates of the center.
The length of the radius.
For an ellipse, state:
The coordinates of the center.
The coordinates of the vertices.
The coordinates of the foci.
For a parabola, state: The coordinates of the vertex.
The coordinates of the focus.
The equations of the directrix line.
For a hyperbola, state: The coordinates of the center.
The coordinates of the vertices.
The coordinates of the foci.
The equations of the asymptotes.
a)
x 2 + 4 y 2 + 6 x + 4 y + 6 = 0 ; what functions would you key into your
calculator to view this conic section?
b)
2 x 2 + 2y 2 + 4 x − 8 y − 6 = 0
c)
9 y 2 − 4 x 2 + 36 y + 16 x − 16 = 0
d)
y 2 − 4 x = −8 ; what functions would you key into your calculator
to view this conic section?
a)
Find the equation of a parabola with a focus at (3,5) and vertex at (3,9).
b)
Find the equation of an ellipse with vertices at (1,0) and (1,8) with a minor
axis of length 2.
c)
Find the equation of a hyperbola with vertices at (6,1) and (0,1) and
focus at (8,1).
d)
Find an equation of a circle where P(1,3) and Q(4,7) are the endpoints of
a diameter.
2)
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MTH 175 – Precalculus
3.
Final Exam Review
Monroe Community College
For each equation, (1) identify the type of conic section (parabola, ellipse, or
hyperbola); (2) sketch the graph; (3) find the center, foci, and vertices, if it is an
ellipse or hyperbola; or find the vertex, focus, and directrix, if it is a parabola; (4)
write the equations of the asymptotes if it is a hyperbola.
a)
c)
( x − 1)2 ( y + 2)2
−
=1
16
9
( x + 3)
2
b)
( x − 2)2 ( y + 5)2
+
=1
4
25
− 8 ( y + 6) = 0
Applications
1)
When a company spends x hundreds of dollars on advertising it realizes a profit.
P(x) = 230 + 20x – 0.25x2
What expenditure gives the maximum profit? What is the maximum profit?
2)
From firetower A, a fire with a bearing N 78º E is sighted. The same fire is sighted
from tower B at N 51º W. Tower B is 70 miles east of tower A. How far is it from
tower A to the fire?
3)
The following graph shows a student’s distance from home during his morning
commute.
a)
Find f −1 (33), and explain its meaning in words. b)
Sketch the graph of f −1 (x).
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