Math 106 Study Guide for Chapter 1 & 2
1. Find the equation of the line containing (-2,3) and perpendicular to the line passing through
Is the line
parallel to the line you found? Explain.
and
.
2. A driver going down a straight highway is traveling at 60 ft/sec (about 41 mph) on cruise control, and begins to
accelerate at a rate of 5.2 ft/sec2. The final velocity of the car is given by
where
is the
velocity at time .
(a) Interpret the meaning of the slope and the -intercept in this context.
(b) Determine the velocity of the car after 9.4 seconds.
(c) If the car is traveling at 100 ft/sec, for how long did it accelerate?
3. For which of the following can y be expressed as a function of x. Explain.
(a)
(b)
(c)
(d)
x
-6
-4
-2
-4
y
2
4
6
10
x
-3
-2
-1
1
y
9
5
3
3
4. Determine the domain of each function. State your answer in interval and set notation.
(a)
√
(b)
√
(c) f ( x) 
1 x
1 | x |
(d) h( x) 
2
x 2  3x  10
5. Sketch a graph of the following piecewise-defined function. State its domain and range.
{
6. Find the average rate of change
of the function
.
7. Determine whether each function is even, odd, or neither.
(a) g ( x)  3x  0.01
(b)
f (t )  t 2  46
(b) (c) F ( x) | x  208 |
(d)
q ( x) 
x
1 | x |
between the points
and
8. Determine the intervals over which the function graphed below is increasing and decreasing.
13. State the domain and range for the
graphed below. Then state the intervals where is increasing or
decreasing and intervals where is positive or negative. Assume all endpoints have integer values.
14. From the text, p. 199, #73 (Reading a graph and operations with functions).
Use the graphs below to answer the next two questions.
15. Using the graph of
and , find
16. Using the graph of
and , find
.
.
17. Describe, in order, the transformations used to go from
18. Let f ( x) 
1
and
x2
to
.
. Find:
(a) ( f  g )(1)
(b) ( f
(c) ( f  g )( x)
(d) Domain of
(e) ( f  g )( x)
(f) Domain of ( f  g )( x) in both interval and set notation
g )(2)
in both interval and set notation
f 
 ( x)
g
(h) Domain of 
f 
 ( x) in both interval and set notation
g
(i) 
g
 ( x)
f 
(j) Domain of 
(k)
(l) Domain of
(g) 
g
 ( x) in both interval and set notation
f 
19. Given f ( x)  x 4  8x 2  15 and g ( x) 
in both interval and set notation
x , determine the following. Simplify your answers.
(a)
(b) Domain of
in both interval and set notation
(c)
(d) Domain of
in both interval and set notation
20. Given
21. Simplify
and
into the form
determine
and state its domain in interval notation.
.
22. Solve 3 5  7d  9  15 . State your answer in interval notation.
23. Solve x 4  16  17 x 2 .
24. Solve
x  7  2x  1 .
25. Solve
2x
36
x
 2

x3 x 9 x3.
26. Find the equation of the line passing through the two points: (10,-3), (-8,6).
forms.
Point-Slope form: ____________________________
Slope-Intercept form: ________________________
Standard form: ________________________
Give answer in the following
27. Are the given lines parallel, perpendicular, or neither?
L1: (0,1), (10,5)
L2: 3x + 5y = 15
28. Find the x- and y-intercepts of the line 2x – 5y = 15.
29. Determine the slope and equation of each given line.
a.
b.
30. Find the slope and the x- and y- intercepts of the line 3x – 6y – 18 = 0. Graph the line.
31. A sales person receives a monthly salary of $3000 plus a commission of 5% of sales. Write a linear
equation for the salesperson’s monthly wage W as a function of the monthly sales S. Will a sales person
make no money if they do not make any sales? Explain. How much does a person need to sell in order to
make $5000 in a month?
32. Suppose that the value of a new car has a linear depreciation in value. If the value of a new car is $18,000
and its worth is only $10,000 after 4 years, then how much is the car worth after 8 years? When will the
car have no value at all? Write a linear equation for the value of the car as a function of its age in years,
and use it to answer the questions.
33. Identify and graph the toolbox (parent) function for f(x)= 3 4  x  5 .
Describe the transformation that occurs in the function, and then graph the function.
Identify any intercepts.
34. Identify and graph the toolbox (parent) function for f(x)= 
1
x  2 1.
3
Describe the transformation that occurs in the function, and then graph the function.
Identify any intercepts.
35. Determine the domain of each of the following functions. Give answers in interval notation.
(a) f(x) =
5x
x 7
(b) g(x) =
36. Find the following difference quotients. Simplify completely.
3 x
x  5x
2
f (3  h)  f (3)
,h  0.
h
g ( a  h)  g ( a )
,h  0 .
(b) Let g(x) = 5x – 2x2. Find the difference quotient
h
(a) Let f(x) = x2 – 4x + 7. Find the difference quotient
37. Use the graphs of f and g to evaluate the following:.
y = g(x)
y = f(x)
38. Given the graph of f(x) on the axes below, find the following. Use Interval notation.
(a) Find the domain of f(x): ___________________________
(b) Find the range of f(x): _____________________________
(c) Over which interval(s) is the function increasing? ___________________________
(d) Over which interval(s) is the function decreasing? ___________________________
(e) Over which interval(s) is the function constant? _____________________________
(f) f(4) = _______ f(0) = _______
f(7) = _______
f(1) = _______
f(2) = _______
(g) Graph the function y = f(x + 6) + 3 on the same axes; label the points of the transformed graph.
Describe the transformation verbally
39. Given the graph of h(x), graph the following functions. Label the points that you graph clearly.
b) y = h(x  2)
(a) y = 2h(x)
h(x)
h(x)
(3, 1)
(3, 1)
(2, 2)
(2, 2)
40. Given f x   x  3 and g x   2 x 2  5 , determine the domain of g  f x  . State your answer in
interval notation.
41. Use the given functions to find the following information. If it does not exist, write DNE (does not exist).
Give the domain in interval notation. Simplify your answers completely.
Graph of 𝑓 𝑥
4
(-6,4)
=
2
(-4,0)
(3,1)
(0,1)
y
(4,0)
(-1,0)
-5
=
5
-2
(-3,-2)
(7,-3)
=
-4
=
Graph of 𝑔 𝑥
(-7,4)
4
=
(-4,2)
2
(-7,1)
-5
(0,0)
5
(2,-1)
-2
-4
(5,-1)
Domain of
: