# Finding domain, compound interest and log

Please see the attachment.

1) Find the domain of the following:

a) g(x) = 5^x

b) g(x) = ln(x-4)

c) g(t) = log(t+2)

d) f(t) = 5.5e^t

2) Describe the transformations on the following graph of f(x) = e^x. State the placement of the horizontal asymptote and y-intercept after the transformation. For example, left 1 or rotated about the y-axis are descriptions.

a) g(x) = e^x - 5

b) h(x) = -e^x

3) Describe the transformations on the following graph of f(x) = log(x). State the placement of the vertical asymptote and x-intercept after the transformation. For example, left 1 or stretched vertically by a factor of 2 are descriptions.

a) g(x) = log(x-3)

b) g(x) = -log(x)

4) The formula for calculating the amount of money returned for an initial deposit into a bank account or CD (certificate of deposit) is given by

A = P (1 + (r/n))^nt

A is the amount of the return.

P is the principal amount initially deposited.

r is the annual interest rate (expressed as a decimal).

n is the number of compound periods in one year.

t is the number of years.

Carry all calculations to six decimals on each intermediate step, then round the final answer to the nearest cent.

Suppose you deposit $3,000 for 9 years at a rate of 6%.

a) Calculate the return (A) if the bank compounds annually (n = 1). Round your answer to the hundredth's place.

b) Calculate the return (A) if the bank compounds quarterly (n = 4). Round your answer to the hundredth's place.

c) Does compounding annually or quarterly yield more interest? Explain why.

d) If a bank compounds continuously, then the formula used is A = Pe^rt

where e is a constant and equals approximately 2.7183.

Calculate A with continuous compounding. Round your answer to the hundredth's place.

e) A commonly asked question is, “How long will it take to double my money?” At 6% interest rate and continuous compounding, what is the answer? Round your answer to the hundredth's place.

5) Suppose that the function P = 11 + 44ln x represents the percentage of inbound e-mail in the U.S. that is considered spam, where x is the number of years after 2002.

a) Use this model to approximate the percentage of spam in the year 2006 to the nearest tenth of a percent.

b) Use this model to determine in how many years it will take for the percent of spam to reach 85% provided that law enforcement regarding spammers does not change.

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