Download MATH 5a EXTRA PRACTICE FOR EXAM 2 1. Solve the following

MATH 5a
EXTRA PRACTICE FOR EXAM 2
1. Solve the following inequalities. Write your answers in interval notation.
3
2
a. x − 4x − 5x > 0
x2 − 2
b.
≤0
x−7
2. A graph is defined by the equation x2 + 2xy − y 2 = 4. Find the x- and y-intercepts (if
any).
3. The following table shows the Super Bowl winners for the years 2002–2010. Is the year
a function of the Super Bowl winner? Briefly justify your answer. Be specific!
Year Super Bowl Winner
2002
New England
2003
Tampa Bay
2004
New England
2005
New England
2006
Pittsburgh
2007
Indianapolis
2008
New York Giants
2009
Pittsburgh
2010
New Orleans
4. Find the equation of the vertical line passing through the point (−2, 4).
5. Find the equation of the line that is perpendicular to the line 6x − 2y = 1 and passes
through the point (2, −1). Graph both lines on the axes below.
6. The graph of a function f (x) is shown below. Use it to answer the following questions.
f (x)
(a) For what value(s) of x, if any, is f (x) = 0?
(b) For what value(s) of x, if any, is f (x) = −4?
(c) Find the domain of f (x). Write your answer in interval notation.
(d) Find the range of f (x). Write your answer in interval notation.
(e) On what interval(s) is the graph of f (x) increasing? Write your answer in interval
notation.
(f) On what interval(s) is the graph of f (x) positive? Write your answer in interval
notation.
(g) Find the average rate of change of f (x) from x = −3 to x = −1.
(h) On the axes below, sketch the graph of y = − 21 f (x − 2) + 1.
7. Let f (x) =
1
f (x + h) − f (x)
. Find
and simplify as much possible.
3x − 2
h
8. Find the domain of the following functions. In part (a) write your answer using interval
notation. In part (b) you may use any notation for your answer, as long as it is clear.
√
x+2
(b) f (x) = 3
(a) f (x) = 3 − 17x
x − 5x
9. Let f (x) =
4x − 1
.
x + 3x + 1
2
(a) Find the domain of f (x).
(b) For what value(s) of x is f (x) = 0?
x + 4y = 1





1
if x < −1
x2
.
10. Sketch the graph of the following function on the axes below: f (x) =
3

x
,
if
−1
≤
x
≤
1


√
x, if x > 1
y = 4x + 5
11. A study at an army training base found that the speed y at which a soldier can run
when he is carrying equipment on his back is related to the weight x of the equipment
by a linear equation. For example, a soldier carrying 20 pounds of equipment can run
at a speed of 7 miles per, hour while a soldier carrying 40 pounds of equipment can
run at a speed of 2 miles per hour.
(a) Find the linear equation relating x and y.
(b) What is the physical interpretation of the slope in this problem? Be very specific.
(c) What is the x-intercept of the line you just found? What is its physical interpretation in this problem?





1
if x < −1
x+2
12. Sketch the graph of the following function on the axes below: f (x) = 1
.


|x − 1|, if −1 ≤ x ≤ 2

2
4 − x,
if x > 2
13. Determine whether each of the following statements is true or false. If a statement is
true, briefly explain why. If a statement is false, else give an example (in the form of
a formula or a picture) that shows that it’s false.
(a) The graph of a function can have two y-intercepts.
(b) Suppose that f (x) is a function whose graph is not a straight line. If the average
rate of change of a function f (x) between x = a and x = b is positive, then f (x)
must be increasing on the interval (a, b).