Math 1090-002
25 November 2011
Exam III review
The following problems are representative of those you will encounter on the exam.
NOTE: You are responsible for all material from the homework assignments for these sections.
Section 3.5
1.
f (x) =
−6
+2
x2
(a) What is the domain of f (x)?
(b) Find all vertical asymptotes.
(c) Does f (x) have a horizontal asymptote? If so, what is it?
1
(d) Find all x-intercepts of f (x).
(e) Does f (x) have a y-intercept? If so, what is it?
(f ) Plot f (x) on the axes below.
2
2.
f (x) =
3x + 2
2−x
(a) What is the domain of f (x)?
(b) Find all vertical asymptotes.
(c) Does f (x) have a horizontal asymptote? If so, what is it?
3
(d) Find all x-intercepts of f (x).
(e) Does f (x) have a y-intercept? If so, what is it?
(f ) Plot f (x) on the axes below.
4
3.
f (x) =
x2 + 2x − 3
x+1
(a) What is the domain of f (x)?
(b) Find all vertical asymptotes.
(c) Does f (x) have a horizontal asymptote? If so, what is it?
5
(d) Find all x-intercepts of f (x).
(e) Does f (x) have a y-intercept? If so, what is it?
(f ) Plot f (x) on the axes below.
6
Section 3.7
Given the pairs of functions f (x) and g(x), perform the indicated operations. Please simplify where possible,
e.g. combine all like terms, find common denominators, etc.
1.
f (x) =
√
g(x) = x2 − x + 1
x+5
(a) (f − g)(x)
(b) (f g)(x)
(c) (g ◦ f )(x)
(d) (f ◦ g)(x)
7
2.
f (x) =
2x
x−1
g(x) =
g
(a)
(x)
f
(b) (f ◦ g)(x)
(c) (g ◦ g)(x)
(d) (f + g)(x)
8
−1
x
Section 4.1
For each function f (x) below, compute the inverse function f −1 (x).
1.
f (x) =
2.
f (x) =
5x
1−x
q
3
1
x
9
3.
f (x) = −x2 + 2, given x ≤ 0
4.
√
g(x) = 3 x + 1
What is the domain of g −1 (x)?
10
Sections 4.2 + 4.3
Rewrite each logarithmic equation below as its equivalent exponential equation using the equivalence
↔
y = loga x
1.
log2 32 = 5
2.
log7
1
49
ay = x.
= −2
Rewrite each exponential equation below as its equivalent logarithmic equation, again using the equivalence
ay = x
3.
ex = 2
4.
10−1 = 0.1
↔
11
y = loga x.
5. Define the function f (x) to be
f (x) = − log(x − 1).
(a) Sketch the graph of f (x). Label the x-intercept.
(b) What is the domain of f (x)?
12
(c) Compute f −1 (x).
(d) Sketch the graph of f −1 (x). Label the y-intercept.
13
6. Define the function f (x) to be
f (x) = ex − 2.
(a) Sketch the graph of f (x). Label the y-intercept.
(b) Compute f −1 (x).
14
(c) Sketch the graph of f −1 (x). Label the x-intercept.
(d) What is the domain of f −1 (x)?
15
Section 4.4
Given
logb x = 3,
logb y = − 31 ,
logb z = 4
evaluate the following expressions.
1.

1
logb (xyz) 2 
2.
logb

xy z3
− logb (x2 z)
16
Section 4.5
Solve the following equations.
1.
xex = 2x2 ex
2.
5x 53x = 510
3.
3x+1 + 2 = 4(3x+1 ) − 7
2
17
2
4.
eln(x
+x)
5.
log2 (x2 ) − log2 (x + 5) = 2
6.
1
−3=0
log5 (x + 2) + log5
x
= 90
18
Properties of logarithms
loga 1 = 0
loga a = 1
loga ax = x
aloga x = x, for all x > 0
loga (mn) = loga m + loga n
loga (mn ) = n(loga m)
loga ( m
n ) = loga m − loga n
loga ( n1 ) = − loga n
Change of base formula
logb x =
loga x
loga b
Properties of exponents
a0 = 1
am an = am+n
(am )n = amn
am
an
= am−n
= a−n
√
1
an = n a
1
an
19