Linear vs. Exponential Functions

What is the smallest, positive integer value of x such that the value of f(x) = 2^x + 3 exceeds the value of g(x) = 50x + 5?

• 8

• 9

• 10

• 11

Two functions are listed below:

f(x) = 100 + 20x

g(x) = 20(1.5)^x

Which statement is true when x = 6?

• The value of f(x) exceeds the value of g(x) by about 8.

• The value of g(x) exceeds the value of f(x) by about 8.

• The value of f(x) exceeds the value of g(x) by about 48.

• The value of g(x) exceeds the value of f(x) by about 48.

Two functions are shown below.

f(x) = 350x + 400

g(x) = 200(1.35)^x

Which statement best describes the two functions?

• f(x) always exceeds g(x)

• f(x) is always less than g(x)

• f(x) > g(x) for whole numbers less than 10

• f(x) < g(x) for whole numbers less than 10

Two functions are graphed.

What is the first and last integer value of x in which the value of f(x) exceeds the value of g(x)?

• -6 and 3

• -6 and 4

• -5 and 3

• -5 and 4

Rodney offer his sister two different pay options to help him mow lawns.

-Option 1: $7 for each hour she works

-Option 2: $0.25 for the first hour she works, and then double the amount of money she earns each additional hour she works.

When would Option 2 first have a higher hourly pay?

• 5 hours

• 6 hours

• 8 hours

• 9 hours

Two functions are shown below.

f(x) = 2^x

g(x) = 16 - 2x

For what positive integer, x, will the value of f(x) first exceed the value of g(x)?

• 3

• 4

• 5

• 6

At how many points does the graph f(x) = 3^x intersect the graph of g(x) = 10x - 3?

• 0

• 1

• 2

• 3

Martha currently earns $32,500 each year. She has been offered two different options for a pay raise.

-The first option is a $750 increase in salary at the start of each year.

-The second option is a 2% increase in salary at the start of each year.

Which statement is true?

• The first option will always result in a higher salary.

• The second option will result in a higher salary at the start of the 2nd year.

• The second option will result in a higher salary at the start of the 16th year.

• There is no difference in the salaries since $750 is approximately 2% of her current salary.

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Two functions are shown below.

g(x) = 12x - 4

h(x) = 1.5(1.11)^x

What is the smallest positive integer value of x such that the value of h(x) exceeds the value of g(x)?

• 49

• 59

• 69

• 79

A scientist is observing the size of a sample of bacteria. The function f(t) = 1000(0.995)^t models the size of the sample t hours after the scientist began his observations. Which statement is true about the sample?

• The sample is growing at a rate of 99.5% per hour.

• The sample is decaying at a rate of 99.5% per hour.

• The sample is growing at a rate of 0.5% per hour.

• The sample is decaying at a rate of 0.5% per hour.