1
c
Kathryn
Bollinger, January 28, 2014
Week-In-Review #2 (1.5 and 3.1)
1. Are the following one-to-one functions?
g(x)
f(x)
f(x)
f(x)
5
4
4
4
2
−5
5
−4
−2
2
−4
4
−4
−2
4
−4
−2
−4
−5
2. Find the domain of the following functions. Write your answer using interval notation.
(a) f (x) = log8 (5x − 9)
(b) g(x) =
3x2
log(x + 2)
3. Rewrite the following in logarithmic form:
(a) 25 = 32
(b) e0 = 1
4. Write the following as a single logarithm:
1
(logb x − logb y + 2 logb z)
3
2
−4
4
2
c
Kathryn
Bollinger, January 28, 2014
5. Write the following as a sum, difference, and/or multiple of logarithms: ln
6. If logb 3 = 1.578 and logb 7 = 2.638, evaluate logb
3x(x + 1)
(2x + 1)2
49b
.
3
7. Solve the following for x algebraically. Give your solution as an EXACT answer.
(a) 2e2x + 5ex = 3
(b) 3e4x − 1 = 7
(c) 2 · 72x = 215
c
Kathryn
Bollinger, January 28, 2014
(d) ln (log3 (2x + 7)) = 0
(e) log (2x + 4) + log (x + 6) = 1
(f) log4 (x + 8) = 2 + log4 (x − 7)
(g) logb x =
1
logb 9 − logb 4 where b > 0 and b 6= 1
2
3
4
c
Kathryn
Bollinger, January 28, 2014
8. How long will it take for your money to triple in an account paying 8% annual interest compounded continuously?
9. Use the given graph of f (x) to answer the following questions.
7
f(x)
1
0
0
1
−5
1
0
0
1
5
1
0
0
1
−7
(a)
(d)
lim f (x) =
(b)
lim f (x) =
(e) lim f (x) =
x→−1−
lim f (x) =
x→−1+
(g) lim f (x) =
x→1
(h) f (−1) =
x→−5
x→−1
(f) lim+ f (x) =
x→0
x→−3+
(c) lim f (x) =
; f (−5) =
(i) List any points of discontinuity and state which condition of continuity is violated at each of
these points.
x−2
, if it exists.
x→2 (x − 2)3
10. Numerically estimate the following limit: lim
x→
f (x)
x→
f (x)
c
Kathryn
Bollinger, January 28, 2014
5


 x+3
, x < −1
0
, x = −1 , algebraically find the following.

 x2 + 1 , x > −1
If a limit does not exist, explain why not.
11. Given f (x) =
(a)
(b)
lim f (x)
x→−1−
lim f (x)
x→−1+
(c) lim f (x)
x→−1
(d) lim f (x)
x→−3
(e) lim f (x)
x→5
12. Algebraically find each of the following, if it exists. If a limit does not exist, explain why not.
2x2 + 5x − 4
x→1
x+2
(a) lim
2x + 3h
h→0 x + 1
(b) lim
x4
x→0 x3 + 2
(c) lim
x2 − 3x − 10
x→5
x−5
(d) lim
c
Kathryn
Bollinger, January 28, 2014
(e) lim
x→−1 x2
(f) lim
x→3
x+2
+ 3x + 2
x−3
x2 − 9
(g) lim
x→−1
√
x+5−2
x+1
1
f (x + h) − f (x)
if f (x) =
h→0
h
x−7
(h) lim
6
7
c
Kathryn
Bollinger, January 28, 2014
x + 2ex
x→0 ex + 1
(i) lim
2x2 − 8
x→2 |x − 2|
(j) lim
(k) If f (x) =

2x2 + 1




 x2 + 1


x+5



x3 + 8
,x<0
, find lim f (x)
x→0
,x≥0
c
Kathryn
Bollinger, January 28, 2014
(l) Given f (x) =
(
2x2 − 8
,x<1
√
, find
−4 x − 2 , x > 1
(i) lim f (x)
x→1
(ii) lim f (x)
x→4
(iii) f (1)
13. Determine the interval(s) where the following functions are continuous.
(a) f (x) =
x+7
(x + 7)(x − 9)
(b) g(x) = x100 + x−1
(c) h(x) = 2ex + 5
8
9
c
Kathryn
Bollinger, January 28, 2014
14. Where is f (x) =

x−6



x+1


 (x − 5)(x + 1)
,x≤4
,x>4
discontinuous?
15. Find the value(s) of a and b, if any exist, that will make g(x) =
continuous on (−∞, ∞).


 5a − x
,x<2
6
,x=2


bx2 + 4 , x > 2