MATH 150 EXAM 1 REVIEW

Domain & Range

100

2x + 1

D: (-∞, ∞)

R: (-∞, ∞)

200

|3x + 2| - 1

D: (-∞, ∞)

R: [-1, ∞ )

300

2x² - 6x - 8

D: (-∞, ∞)

R: [-25/2, ∞)

400

√(4 - x) - 10

D: (-∞, 4]

R: [-10, ∞)

500

-5 / (8 - 2x - x²)

D: (-∞, -4) U (-4, 2) U (2, ∞)

R: (-∞, 0) U (0, ∞)

Combining & Inverse

100

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

200

Find the inverse

f(x) = √(x + 1) / 7

f^-1(x) = 49x² - 1

300

Find the inverse

f(x) = (2x + 3) / (1 - 5x)

f^-1(x) = (x - 3) / (5x + 2)

or

f^-1(x) = (3 - x) / (-5x - 2)

400

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

500

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

Misc.

100

Describe the transformation

log(-2) + x

DNE

200

Describe the transformation:

-(x + 3)² - 4

Reflects over x-axis

Shifts to the left 3

Shifts down 4

300

Describe the transformation:

f(x) = -4 - 2^{-x + 1}

Reflects over x-axis

Reflects over y-axis

Shifts down 4

400

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500

When you can't afford premium:

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Logs & Exp.

100

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

200

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

300

Solve for x:

a) 2^{x^2 - 1} = 6^{1 - x^2}

b) 2log(x) = log2 + log(3x - 4)

a) x = 1, -1

b) x = 2, 4

400

Combine the logarithm

4log(x) - 1/3 (log(x² + 1)) + 2log(x - 1)

log(x⁴(x - 1)² / (3√x² + 1))

500

Expand the logarithm

log√(x² + 4 / (x² + 1)(x³ - 7)²)

1/2[log(x² + 4) - log(x² + 1) - 2log(x³ - 7)]

Compound Formulas

100

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

200

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%

300

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) = n₀2^t/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 * 2^2t/3 or 10 * 2^t/1.5

b) 10(2^35/1.5)

c) t = 4.5 / log(2)

400

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) = n₀e^rt 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) = 18,000e^0.08t

b) 18,000e^0.64

c) t = ln(25/18) / 0.08

500

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) 22/2^5/2

d) t = -1600ln(9/11)/ln(2)