Domain & Range
2x + 1
D: (-∞, ∞)
R: (-∞, ∞)
Find the functions f º ɡ, ɡ º f, f º f, and ɡ º ɡ
f(x) = x2
g(x) = x + 1
(f º g)(x) = (x + 1)2
(g º f)(x) = x2 + 1
(f º f)(x) = x4
(g º g)(x) = x + 2
Describe the transformation
log(-2) + x
DNE
State domain, range, asymptotes, x-int, y-int
f(x) = 2 - (1/3)x
D: (-∞, ∞)
R: (-∞, 2)
x-int: (-(ln(2)/ln(3)), 0)
y-int: (0, 1)
HA: y = 2
VA: None
Brenda invests $4,848 in a savings account with a fixed annual interest rate of 5% compounded 2 times per year. What will the account balance be after 6 years?
$6,520.02
|3x + 2| - 1
D: (-∞, ∞)
R: [-1, ∞ )
Find the inverse
f(x) = √(x + 1) / 7
f-1(x) = 49x2 - 1
Describe the transformation:
-(x + 3)2 - 4
Reflects over x-axis
Shifts to the left 3
Shifts down 4
State domain, range, asymptotes, x-int, y-int
f(x) = log4(16 + x) + 3
D: (-16, ∞)
R: (-∞, ∞)
x-int: (-1023/64, 0)
y-int: (0, 5)
HA: None
VA: x = -16
Adam invests $6,139 in a retirement account with a fixed annual interest rate compounded continuously. After 17 years, the balance reaches $8,624.97. What is the interest rate of the account?
2%
2x2 - 6x - 8
D: (-∞, ∞)
R: [-25/2, ∞)
Find the inverse
f(x) = (2x + 3) / (1 - 5x)
f-1(x) = (x - 3) / (5x + 2)
or
f-1(x) = (3 - x) / (-5x - 2)
Describe the transformation:
f(x) = -4 - 2-x + 1
Reflects over x-axis
Reflects over y-axis
Shifts down 4
Solve for x:
a) 2x^2 - 1 = 61 - x^2
b) 2log(x) = log2 + log(3x - 4)
a) x = 1, -1
b) x = 2, 4
A certain culture of the bacterium Streptococcus A initially has 10 bacteria and is observed to double every 1.5 hours.
a) Find an exponential model n(t) = n02t/a
b) Estimate the number of bacteria after 35 hours
c) After how many hours will the bacteria count reach 10,000
a) n(t) = 10 * 22t/3 or 10 * 2t/1.5
b) 10(235/1.5)
c) t = 4.5 / log(2)
√(4 - x) - 10
D: (-∞, 4]
R: [-10, ∞)
Find f + ɡ, f − ɡ, fg, and f/g
f(x) = 2/x
g(x) = 4 / (x + 4)
f + g = (6x + 8) / (x2 + 4x)
f - g = (-2x + 8) / (x2 + 4x)
fg = 8 / (x2 + 4x)
f/g = (x + 4) / 2x
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Combine the logarithm
4log(x) - 1/3 (log(x2 + 1)) + 2log(x - 1)
log(x4(x - 1)2 / (3√x2 + 1))
The fox population in a certain region has a relative growth rate of 8% per year. It is estimated that the population in 2013 was 18,000.
a) Find a function n(t) = n0ert that models the population t years after 2013.
b) Estimate the fox population in the year 2021
c) After how many years will the fox population reach 25,000
a) n(t) = 18,000e0.08t
b) 18,000e0.64
c) t = ln(25/18) / 0.08
-5 / (8 - 2x - x2)
D: (-∞, -4) U (-4, 2) U (2, ∞)
R: (-∞, 0) U (0, ∞)
Find the functions f º ɡ, ɡ º f, f º f, and ɡ º ɡ
f(x) = x / x + 1
g(x) = 2x - 1
(f º g)(x) = 2x - 1 / 2x
(g º f)(x) = x - 1 / x + 1
(f º f)(x) = x / 2x + 1
(g º g)(x) = 4x - 3
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Expand the logarithm
log√(x2 + 4 / (x2 + 1)(x3 - 7)2)
1/2[log(x2 + 4) - log(x2 + 1) - 2log(x3 - 7)]
The half-life of radium-226 is 1600 years. Suppose we have 22-mg sample.
a) Find a function m(t) = m02-t/h that models the mass remaining after t years.
b) Find a function m(t) = m0e-rt that models the mass remaining after t years.
c) How much of the sample will remain after 4000 years?
d) After how many years will only 18 mg of the sample remain?
a) m(t) = 22(2-t/1600)
b) m(t) = 22(e(-ln(2)/1600)t)
c) 22/25/2
d) t = -1600ln(9/11)/ln(2)