Review over Chapter 4
For the following graphs, answer the following
a) Is it a function?
b) Find the domain.
c) Find the range.
d) Does it have an inverse function?
1)
2)
6
8
5
7
4
6
3
5
2
1
4
0
3
-6 -5 -4 -3 -2 -1
-1 0
1
2
3
4
5
6
2
-2
1
-3
0
-4
-5
-5
-4
-3
-2
-6
-1 0
-1
1
2
-2
Are f(x) and g(x) inverse functions? Show your justification:
3)
f(x) = 4x – 5 and g(x) = 0.5x + 1.25
4)
f(x) =
2
x−5
and g(x) =
2
x
+5
Evaluate the following. Round to four decimal places:
5a)€ log(23.2)
5c) log(– 0.63)
€
–5
5b)
5d)
ln(0.0365)
5•ln( 10 )
6a)
( 21 )
6b)
e
6c)
(8.326) – 2/3
6d)
€
(1.81) –
3
29
€
€ Find the domain of the following functions:
7)
f(x) = 4x – 3
8) €j(x) = log6(x – 8)
9)
g(x) =
€
3x + 2
( )
1
2
10)
f(x) = ln(– x2 – 4x + 12)
3
4
5
Write the following as an exponential expression. Do Not Simplify:
11a) r = log3/5(10)
11b) 7 = ln(q)
Write the following as an logarithmic expression. Do Not Simplify:
12a)
–2
( )
5
6
36
25
=
12b) ex = 0.085
Evaluate the following using the change of base formula. Round to
four decimal places:
€
€ loga (M)
logb(M) =
loga (b)
13a) log3(5)
13c) log
10
13b) log1/5(8.4)
(0.62)
13d) logπ( 3 17 )
€
Find the inverse of each function if it exists:
€
€
1
14) f(x) = – x + 4
15) g(x) =
3
Solve the following:
e4x – 1 = (e2)3x
€ 2x + 1
18) 9
= 243x – 2
16)
2
x−3
3x + 2
( )
1
17)
= 16
8
€ 3x
19) 2 + 1 = 3x – 2
Solve the following. Round to the nearest
thousandth if needed:
€
20)
3(4)2x + 1 + 11 = 65
22)
log14(4 – 3x) + log14(1 – 2x) = 1
23)
ln(5x + 3) + ln(2x – 1) = ln(9x2 + x + 1)
21)
6Ln(x) – 8 = 15
Write the expression as a single logarithm:
24) 4log5(x + 7) – log5(3x – 2) + 2log5(9 – 4x)
25) 3ln(2x + 9) + ln(14x – 5) – 2ln(x + 7)
Write the expression as a sum and/or difference of logarithms.
Express powers as factors.
26)
€
log
y 5 (x−8)
( (3x−5) )
4
27)
€
ln
(x−15)6
(x
x+3
)
Solve the following (round to two decimal places):
28) Find the amount of time it would take for $3200 to double is it is
invested at:
a)
8.45% compounded quarterly.
b)
8.45% compounded continuously.
29)
Find the amount that Juanita should invest in an account paying
6.15% annual interest so that after 20 years, she will have $50,000
if the interest is compounded:
a) monthly
b) continuously
30)
A colony of bacteria increases according to the law of uninhibited
growth. The number of bacteria doubles in size every nine hours.
a)
How long will it take for the number of bacteria to be 16 times
its original size?
b)
How long will it take for the number of bacteria to triple?
31)
It is projected that t years from now, the population of a certain
country will be P(t) = 72e0.023t million people. Approximately when will
the population reach 200 million people?
32)
The amount of a sample of radioactive substance remaining after t
years is given by Q(t) = Qoe – 0.015t.
a)
What is the half-life?
b)
At the end of 70 years, 300 grams of the substance remained.
How many grams were initially present?
Answers:
1a)
Yes
1b) Domain: (– ∞, ∞)
1c) Range: (– ∞, ∞)
1d)
Yes
2a)
Yes
2b) Domain: [– 4, 4)
2c) Range: [0, 4]
2d)
No
3)
5d)
6d)
No 4) Yes
≈ 5.7565 6a)
≈ 0.0410 7)
11a)
( )
3
5
r
= 10
13a) ≈ 1.4650
5a) ≈ 1.3655 5b) ≈ – 3.3104 5c) Undefined
32 6b) ≈ 5.6522 6c) ≈ 0.2434
(– ∞, ∞) 8) (8, ∞) 9) (– ∞, ∞) 10) (– 6, 2)
11b) e7 = q
12a) log5/6(
13b) ≈ – 1.3223
18) {12}
)=–2
13c) ≈ – 0.4152
12b) ln(0.085) = x
13d) ≈ 0.8250
2
14) f – 1(x) = – 3x + 12
€
36
25
15) g – 1(x) =
+ 3 16) {– ½}
17) {–
x
€
19) {– 2.9469} 20) {0.542} 21) {46.216} 22) {– 2/3}
23) {2} 24) log5
(
(x+7) 4 (9−4x)2
€
3x−2
)
25) ln
26) 5log(y) + log(x – 8) – 4log(3x – 5)
28a) ≈ 8.29
€ years
(
(2x+9)3 (14x−5)
(x+7)2
)
10
9
}
€
27) 6ln(x – 15) – ln(x) –
1
ln(x
2
+ 3)
28b) ≈ 8.20 years 29a) $14,660.61
€
29b) $14614.63 30a) 36 hours 30b) ≈ 14.26 hours 31) ≈ 44.42 years
€
32a) ≈ 46.21 years 32b) ≈ 857.30 grams