Q: Suppose you are given two options: (a) receive $$0.1$$ million dollars right now, or (b) receive $$0.01$$ million dollars that doubles every day for $$1$$ month (assume $$30$$ days). Which option gives you more money after $$30$$ days?

Option (b) gives more money because $$0.01 \times 2$$^{30}$$ million is much greater than 0.1$ million.

Q: A medication's concentration in the bloodstream decreases by half every $$4$$ hours. If the initial concentration is $$80 mg/L$$, what is the concentration after $$12$$ hours?

$$80 \times (0.5)^3$ = 10 mg/L$$

Question 1

Suppose you are given two options: (a) receive $$0.1$$ million dollars right now, or (b) receive $$0.01$$ million dollars that doubles every day for $$1$$ month (assume $$30$$ days). Which option gives you more money after $$30$$ days?

Accepted Answer

Option (b) gives more money because $$0.01 	imes 2$$^{30}$$ million is much greater than 0.1$ million.

Question 2

Given the exponential function $$f(x) = C a^x$$, which parameter represents the initial value?

Accepted Answer

$$C$$

Question 3

What is the horizontal asymptote of the exponential function $$f(x) = 2^x$$?

Accepted Answer

$$y = 0$$

Question 4

If $$a \u003e 1$$, what is the domain and range of the exponential function $$f(x) = a^x$$?

Accepted Answer

Domain: $$(-\infty, \infty)$$, Range: $$(0, \infty)$$

Question 5

Which of the following best describes the graph of $$f(x) = 0.5^x$$?

Accepted Answer

It decreases as $$x$$ increases.

Question 6

A population of bacteria starts at $$100$$ and doubles every $$3$$ hours. What is the population after $$9$$ hours?

Accepted Answer

$$100 	imes 2^3$ = 800$$

Question 7

Which property is NOT true for all exponential functions $$f(x) = a^x$$ with $$a \u003e 0$$ and $$a 
eq 1$$?

Accepted Answer

The function is always increasing.

Question 8

If $$f(x) = 3^x$$ and $$g(x) = 2^x$$, which function grows faster as $$x$$ increases?

Accepted Answer

$$f(x)$$ grows faster because $$3 \u003e 2.$$

Question 9

Which of the following is the correct formula for an exponential function with initial value $$C$$ and growth factor $$a$$?

Accepted Answer

$$f(x) = C a^x$$

Question 10

A medication's concentration in the bloodstream decreases by half every $$4$$ hours. If the initial concentration is $$80 mg/L$$, what is the concentration after $$12$$ hours?

Accepted Answer

$$80 	imes (0.5)^3$ = 10 mg/L$$

Question 11

Which of the following statements is true about the function $$f(x) = a^x$$ when $$0 \u003c a \u003c 1$$?

Accepted Answer

The function decreases as $$x$$ increases.

Question 12

If $$f(x) = 2^x$$ and $$f(3) = 8$$, what is $$f(-2)$$?

Accepted Answer

$$2^{-2} = 0.25$

Question 13

Which of the following real-world scenarios is best modeled by an exponential function?

Accepted Answer

The depreciation of a car's value by a fixed percentage each year.

Question 14

If $$f(x) = 2^x$$ and $$g(x) = 2$$^{-x}$$, which statement is true?

Accepted Answer

$$f(x)$$ increases and $$g(x)$$ decreases as $$x$$ increases.

Question 15

A scientist is tracking the growth of a cell culture that triples every $$2$$ hours. If the initial count is $$500$$ cells, what is the count after $$6$$ hours?

Accepted Answer

$$500 	imes 3^3$ = 13,500$$

Question 16

Which of the following is NOT a property of exponential functions?

Accepted Answer

They have a vertical asymptote at $$x = 0.$$

Question 17

If $$f(x) = 5 \cdot 2^x$$, what is the value of $$x$$ when $$f(x) = 40$$?

Accepted Answer

$$x = 3$$

Question 18

Which graph best represents the function $$f(x) = 2^x$$?

Accepted Answer

A curve that increases rapidly for positive $$x$$ and approaches $$y = 0$$ for negative $$x.$$

Question 19

A radioactive substance decays so that its mass halves every $$5$$ years. If the initial mass is $$160$$ grams, what is the mass after $$15$$ years?

Accepted Answer

$$160 	imes (0.5)^3$ = 20$$ grams

Exponential Functions: Definitions, Properties, and Graphs

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