Chapter 6 Review
#1-2
a) (𝑓 ∘ 𝑔)(𝑥)
b) The domain of (𝑓 ∘ 𝑔)(𝑥)
1) f(x) = x2 + 2x + 3
g(x) = 4x – 1
1a) (𝑓 ∘ 𝑔)(𝑥) = (4x-1)2 + 2(4x – 1) + 3
= (4x - 1)(4x – 1) + 2(4x – 1) + 3
= 16x2 – 4x – 4x + 1 + 8x – 2 + 3
= 16x2 + 2
Answer: (𝒇 ∘ 𝒈)(𝒙) = 2(8x2 + 1)
3
2) 𝑓(𝑥) = 𝑥
1b) Since the domain of both f and g is all real
numbers so is the domain of (𝑓 ∘ 𝑔)(𝑥)
Answer: domain (−∞, ∞)
1
𝑔(𝑥) = 𝑥−4
2a) (𝑓 ∘ 𝑔)(𝑥) =
3
1
𝑥−4
=3÷
1
𝑥−4
= 3(𝑥 − 4)
Answer: (𝒇 ∘ 𝒈)(𝒙) = 3x - 12
2b) 4 needs to be excluded from the domain, as
the domain of g(x) is all real numbers except 4. No
additional numbers need to be excluded from the
domain as the answer to 2a) is a polynomial.
Answer: domain all real numbers except 4
#3-4: Determine which of the functions are one to one. If a function is one to one find its inverse.
3) f = { (0,1) (1,4) (2,0) (3,5)}
3) f is one to one (since all y’s are different)
f-1 = { (1,0) (4,1) (0,2) (5,3)}
4) g = { (3,4) (4,2) (-3,4) (1,-5) (8,7)}
4) g is not one to one as there are two points
with the same y value.
No need to find g inverse as it would not be a
function.
#5 – 6: The graph of a one to one f function is given. Draw the graph of the inverse function f-1
5) Just switch (x,y) and draw same shape.
y
18
16
14
12
10
8
6
4
2
(16,2)
-19
-18
-17
-16
-15
-14
-13
-12
-11
-10
-9
-8
-7
-6
-5
-4
-3
-2
(1/4,1/2)
-1
(-2,-1)
x
(2,1)
1
2
3
4
5
6
7
8
9
10
11
(-1/4,-1/2)
-2
(-16,-2)
-4
-6
-8
-10
-12
-14
-16
-18
6) Just switch (x,y) and draw same shape.
y
18
16
14
12
10
(12,9)
8
6
4
(8,4)
2
-19
-18
-17
-16
-15
-14
-13
-12
-11
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
(0,0)
-2
-4
-6
-8
-10
-12
-14
-16
-18
1
2
3
(4,1)
4
x
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
12
13
14
15
16
17
18
19
#7-8: Each of the following functions is one to one. Perform the following:
a) Find the inverse of each function, and express it using appropriate notation.
b) Check your answer by showing that (𝑓 ∘ 𝑓 −1 )(𝑥) = x and (𝑓 −1 ∘ 𝑓)(𝑥) = x
7) f(x) = 2x + 9
7a) f(x) = 2x + 9
y= 2x + 9
x = 2y + 9
x – 9 = 2y
𝑥−9
=𝑦
2
-1
Answer: f (x) =
𝒙−𝟗
)+9
𝟐
7b) (𝒇 ∘ 𝒇−𝟏 )(𝒙) =𝟐 (
=x–9+9
=x
𝒙−𝟗
𝟐
(𝒇−𝟏 ∘ 𝒇)(𝒙)
𝟐𝒙+𝟗−𝟗 𝟐𝒙
= 𝟐
𝟐
=
=𝒙
8) f(x) = x3 – 5
8a) f(x)= x3 – 5
8b)
𝟑
𝟑
y = x3 – 5
(𝒇 ∘ 𝒇−𝟏 )(𝒙) = (√𝒙 + 𝟓) − 𝟓 = 𝒙 + 𝟓 − 𝟓 = 𝒙
3
x=y –5
x + 5 = y3
𝟑
𝟑
(𝒇−𝟏 ∘ 𝒇)(𝒙) = √𝒙𝟑 − 𝟓 + 𝟓 = √𝒙𝟑 = 𝒙
3
3
√𝑥 + 5 = √𝑦 3
3
√𝑥 + 5 = 𝑦
𝟑
Answer: 𝒇−𝟏 (𝒙) = √𝒙 + 𝟓
9) Describe the transformation of the graph of g(x) as compared to the graph of f(x) = 2x.
9a) g(x) = 2x+5 Answer: shift left 5 units
9b) g(x) = 2x-4 Answer: shift right 4 units
9c) g(x) = 2x + 1 Answer: shift up 1 unit
9d) g(x) = 2x - 3 Answer: shift down 3 units
9e) g(x) = -2x Answer: reflected over x-axis
9f) g(x) = 2-x Answer: reflected over y-axis
9g) g(x) = 2x+3 +5 Answer: shift left 3 units up 5 units
9h) g(x) = 2x-3 - 2 Answer: shift right 3 units down 2 units
9i) g(x) = -2x+1 Answer: reflect over x-axis, shift left 1 unit
9j) g(x) = -2-x Answer: reflect over x-axis and reflect over y-axis
#10-15: Write each side with the same base then solve. Be sure to check your answer.
10) 3x-3 = 27
3x-3 = 33
x–3=3
Answer: x = 6
11) 2x+1 = 16
2x+1 = 24
x+1 = 4
Answer: x = 3
1 𝑥−2
12) (2)
1 𝑥−2
2
( )
1 𝑥−2
(2)
=
1
=8
13
23
1 3
= (2)
x– 2 = 3
Answer: x = 5
1 𝑥−2
13) (3)
1 𝑥−2
3
( )
1 𝑥−2
(3)
=
1
= 81
14
34
1 4
= (3)
x–2=4
Answer: x = 6
14) 𝑒 𝑥+4 ∗ 𝑒 2𝑥 = 𝑒 10
ex+4+2x = e10
e3x+4 = e10
3x + 4 = 10
3x = 6
Answer: x = 2
15) 22𝑥−4 ∗ 23 = 25
22x – 4 + 3 = 25
22x-1 = 25
2x – 1 = 5
2x = 6
Answer: x = 3
16) y = log2(x-3)
16a) Graph the logarithmic functions. First write the equation in exponential form, then create a table
of values and plot the points.
2y = x – 3
2y + 3 = x
x
22 + 3 = 7
21 + 3 = 5
20 + 3 = 4
2-1 + 3 = 3.5
2-2 + 3 = 3.25
y
2
1
0
-1
-2
point
(7,2)
(5,1)
(4,0)
(3.5, -1)
(3.25, -2)
y
9
8
7
6
5
4
3
2
1
x
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
8
9
-1
-2
-3
-4
-5
-6
-7
-8
-9
16b) State the domain of each function.
Set argument > 0
x– 3 > 0
x> 3
Answer: domain (𝟑, ∞)
16c) Describe the transformation the occurs from a common function
Answer: same shape as y = log2x , except shifted 3 units to the right
#17-19: Write the expression as a single logarithm. Write your answer with only positive exponents.
17) 3log2 x - 4log2 y
Make coefficients exponents
= log2 x3 – log2 y4
Use minus to divide rule
𝒙𝟑
Answer: 𝒍𝒐𝒈𝟐 𝒚𝟒
18) 5ln x -2ln y + ln z
Make coefficients exponents, write with minus term last
Ln x5 + ln z – ln y2
Write ln, need fraction because of the minus, only y2 in denominator as it is only log with minus in front.
Answer: 𝒍𝒏
𝒙𝟓 𝒛
𝒚𝟐
19) 2log3x - 4log3y - log3z
Make coefficients exponents
Log3 x2 – log3 y4 – log3 z
Write log3 need fraction because of the minus, both y4 and z belong in denominator as they both have
minus in front.
𝒙𝟐
Answer: 𝒍𝒐𝒈𝟑 𝒚𝟒𝒛
#20-22: Expand into sums and differences of logarithms (express powers as factors).
20) 𝑙𝑜𝑔3
𝑦
𝑥2𝑧
Create 3 logs, one for each variable. The log with the x and z get minus in front as they are in the
denominator. Make exponents into coefficients.
Answer: log3 y - 2log 3x – log3 z
𝑥2𝑦
)
𝑧
21) 𝑙𝑛 (
Create 3 logs, one for each variable. The log with z gets a minus in front as it is in the denominator.
Make exponents into coefficients.
Answer: 2ln(x) + ln(y) – ln(z)
22) log4(x3y2z5)
Create 3 logs, one for each variable. No minus is needed as there is not a fraction. Make exponents into
coefficients.
Answer: 3log4 x + 2log4 y + 5log4 z
#23-26: Solve the exponential equations, round your answer to 2 decimals.
23) 3x = 12
Log 3x = log 12
x(log 3) = log 12
𝑥=
log 12
log 3
Answer x = 2.26
24) 2x = 6
Log 2x = log 6
xlog 2 = log 6
𝑥=
log 6
log 2
Answer: x = 2.58
25) 2x = 5x+4
Log 2x = log 5x+4
x( log 2) = (x + 4)log 5
x(log 2) = x(log 5) + 4log5
x(log 2) – x(log5) = 4log5
x(log 2 – log 5) = 4log 5
4 log 5
𝑥 = log 2−log 5
Answer: x = -7.03
26) 6x-9 = 3x
Log 6x-9 = log 3x
(x-9)log 6 = x(log 3)
x(log 6) – 9log 6 = x(log 3)
-9log 6 = x(log 3) – x(log 6)
-9log 6 = x(log 3 – log 6)
−9 log 6
log 3−log 6
=x
Answer: x = 23.26
#27-36: Solve the logarithmic equations, round to 2 decimals when needed.
27) log3 x = 4
34 = x
Answer: x = 81
28) ln x = 2
Loge x = 2
e2 = x
Answer: x = 7.39
29) log2(x+1) = 5
25 = x + 1
32 = x + 1
Answer: x = 31
30) log4(x-5)=3
43 = x – 5
64 = x – 5
Answer: x = 69
31) log(5x-10) = log(x–2)
5x – 10 = x – 2
5x – x = - 2 + 10
4x = 8
Answer: x = 2
32) ln(2x-4) = ln(3x-10)
2x – 4 = 3x – 10
-4 + 10 = 3x – 2x
Answer: x = 6
33) log2(x+14) – log2(x+6)=1
𝑙𝑜𝑔2
21 =
2
1
=
𝑥+14
𝑥+6
=1
𝑥+14
𝑥+6
𝑥+14
𝑥+6
2(x +6) = 1(x+14)
2x + 12 = x + 14
2x – x = 14 – 12
Answer: x = 2
34) log3 x – log3(x+18) = -1
𝑥
𝑙𝑜𝑔3 𝑥+18 = −1
3−1 =
1
3
𝑥
𝑥+18
𝑥
= 𝑥+18
3x = x + 18
2x = 18
Answer: x = 9
35) log2(x+2)+log2(x+6) = 5
Log2(x+2)(x+6) = 5
Log2(x2 + 8x+ 12) = 5
25 = x2 + 8x+ 12
32 = x2 + 8x + 12
0 = x2 + 8x – 20
0 = (x + 10)(x – 2)
x+ 10 = 0
x = -10 (doesn’t check)
x–2=0
x=2
Answer: x = 2
36) log3 x + log3(x+6) = 3
Log3 x(x+6) = 3
Log3(x2 + 6x) = 3
33 = x2 + 6x
27 = x2 + 6x
0 = x2 + 6x – 27
0 = (x+9)(x-3)
x+9=0
x = -9 (doesn’t check)
Answer: x = 3
x–3=0
x=3
37) How long will it take an initial investment of $2,000 to triple if it is expected to earn 3% interest
compounded continuously? (Round to 1 decimal place) (use formula A = Pert)
A = 6.000 (need to triple $2,000)
P = 2,000 (starting amount)
r = .03
t = solve for
6000 = 2000e.03t
3 = e.03t
Ln 3 = ln e.03t
Ln 3 = .03t(ln e)
Ln 3 = . 03t
ln 3
.03
(ln e = 1)
=𝑡
Answer: 36.6 years
38) How long will it take an initial investment of $5,000 to double if it is expected to earn 2% interest
compounded continuously? (Round to 1 decimal place) (use formula A = Pert)
A = 10,000 (double 5,000)
P = 5,000
r = .02
t = solve for
1000 = 5000e.02t
2 = e.02t
Ln 2 = ln e.02t
Ln 2 = .02t(ln e)
Ln 2 = . 02t
𝑙𝑛2
.02
(ln e = 1)
=𝑡
Answer: 34.7 years
39) The population of a city is expected to double in 40 years. The city currently has 10,000 residents.
How long will it take to get to 50,000 residents? (round final answer to 2 decimals)
(use formula P(t) = P0ekt, round k to 3 decimals)
Use P(40) = 20,000, P0 = 10,000 and t = 40 to solve for k
20000 = 10000ek*40
2 = e40k
Ln 2 = ln e40k
Ln 2 = 40k(ln e)
Ln 2 = 40k
ln 2
40
=𝑘
k = .017
now replace P(t) with 50,000 and k with .017 and solve for t
50000 = 10000e.017t
5 = e.017t
Ln 5 = ln e.017t
Ln 5 = .017t(ln e)
Ln 5 = .017t
ln 5
.017
=𝑡
Answer: 94.67 years
40) The population of a city is expected to grow to 3,300 in 20 years. The city currently has 1,000
residents. How long will it take to get to 5,000 residents? (round final answer to 2 decimals)
(use formula P(t) = P0ekt, round k to 3 decimals)
Use P(20) = 3,300, P0 = 1,000 and t = 20 to solve for k
3300 = 1000ek*20
3.3 = e20k
Ln 3.3 = ln e20k
Ln 3.3 = 20k(ln e)
Ln 3.3 = 20k
ln 3.3
20
=𝑘
k = .060
now replace P(t) with 5,000 and k with .060 and solve for t
5000 = 1000e.060t
5 = e.060t
Ln 5 = ln e.060t
Ln 5 = .060t(ln e)
Ln 5 = .060t
ln 5
.060
=𝑡
Answer: 26.82 years
41) A chemist has a 50-gram sample of a radioactive material. He records the amount of radioactive
material present every week for 6 weeks and obtains the following data
week
Weight
(grams)
0
50
1
40
2
32
3
26
4
22
5
21
41a) Find the appropriate regression equation, and write it in the form P(t) = P0ekt
enter data on calculator and perform expreg
Y = 45.769(.854)t
Now write in desired form
P0 = a = 45.769
k = ln(b) = ln(.854) = -.158
Answer: P(t) = 45.769e-.158t
41b) predict when there will be 10 grams of the material left.
10 = 45.769e-.158t (divide both sides by 45.769)
.218 = e-.158t
Ln .218 = ln e-.158t
Ln .218 = -.158t(ln e)
Ln .218 = -.158t
ln.218
−.158
=𝑡
Answer: 9.64 weeks
6
20