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Math 142 WIR, copyright Angie Allen, Spring 2013
Math 142 Week-in-Review #2 (Sections 1.5 and 3.1)
1. Which, if any, of the following functions have an inverse? Explain why or why not.
a) f (x) =
1
x
b) g(x) = x3 − 2x2 − 5
2. Find the domain of the following functions:
√
6
a) g(x) =
x+2
ln(3x − 5)
b) f (x) =
8 log3 (x − 5)
x2
e x+5
2
Math 142 WIR, copyright Angie Allen, Spring 2013
3. Evaluate the following limits numerically, if they exist. If a limit does not exist, use limits to describe the
way in which it does not exist.
a) lim (x2 − 5)
x→−4
b) lim
x→3
1
x−3
4. If logb 4 = 1.2619 and logb 6 = 1.6310, find logb
16
.
6b2
Math 142 WIR, copyright Angie Allen, Spring 2013
x2 − 7x + 1
, if it exists. If it does not exist, use limits to describe the way in which it does not
x→3 (x − 3)2
5. Evaluate lim
exist.
3
6. Solve the following for x:
a) log3 (x + 5) = log10 100 − log3 (x − 5)
b) 8 1.045x+1 = 24
c) 23 loga 25 − 2 loga 5 = loga x, where a > 0 and a 6= 1.
Math 142 WIR, copyright Angie Allen, Spring 2013
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d) log8 (x + 3) = log8 5 + log8 (x − 1)
x2 + 8x + 15
, if it exists. If it does not exist, use limits to describe the way in which it does
x→−5 x2 + x − 20
7. Evaluate lim
not exist.
x−1
, if it exists. If it does not exist, use limits to describe the way in which it does not
x→1 (x − 1)2
8. Evaluate lim
exist.
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Math 142 WIR, copyright Angie Allen, Spring 2013
9. Consider the function f (x) below and answer the following questions:
a) Find lim f (x), if it exists.
x→−3
y
8
7
6
5
b) Find f (1), if it exists.
4
3
2
c) Find lim f (x), if it exists.
x→1
1
−8 −7 −6 −5 −4 −3 −2 −1
1
−1
−2
−3
−4
−5
−6
−7
−8
d) Find f (2), if it exists.
e) Find lim+ f (x), if it exists.
x→2
f) State, using the definition of continuity, why f
is not continuous at x = −3.
g) State, using the definition of continuity, why f
is not continuous at x = 1.
2 3 4
5
6
7 8
x
6
Math 142 WIR, copyright Angie Allen, Spring 2013
10. Evaluate lim
x→13
exist.
√
x+3−4
, if it exists. If it does not exist, use limits to describe the way in which it does not
x − 13
11. Determine where the function f (x) =
5 ln(x − 3)
√
is continuous.
x+7
12. How long (in years) will it take a piece of equipment to be worth half of its current value if it is depreciating
at a relative rate of 5.6% per year?
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Math 142 WIR, copyright Angie Allen, Spring 2013
x2 − 9
, if it exists. If it does not exist, use limits to describe the way in which it does not exist.
x→3 |x − 3|
13. Evaluate lim
14. For the function f (x) = x−1 , find lim
h→0
the way in which it does not exist.
f (4 + h) − f (4)
, if it exists. If it does not exist, use limits to describe
h
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Math 142 WIR, copyright Angie Allen, Spring 2013
15. Consider the function f (x) below and answer the following questions:
a) Find f (3), if it exists.
b) Find f (−2), if it exists.
c) Find lim f (x), if it exists.
x→0
d) Find lim− f (x), if it exists.
x→3
e) Find lim f (x), if it exists.
x→−2
f) Determine where f (x) is continuous
using the definition of continuity.

10x − 4x2








x+1
2
f (x) =
x + 4x + 3





2


 x −8
6 · 23−x
x < −2
−2 < x ≤ 3
x>3
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Math 142 WIR, copyright Angie Allen, Spring 2013
16. Simplify the following using properties of logarithms:
1
3 log7 x − 2 log7 y + log7 z + 4 log7 75
3
17. Find the value of A such that the function f (x) below is continuous for all real numbers x.
f (x) =
 x
 3e

x < −2
Ax2 + x + 1 x ≥ −2