Q: Which of the following is the correct definition of the number $$e$$?

$$e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n$$

Question 1

Which of the following best describes the domain and range of the exponential function $$f(x) = 3^x$$?

Accepted Answer

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

Question 2

Given the function $$f(x) = 2^x$$, which statement about its graph is TRUE?

Accepted Answer

The graph passes through the point $$(0, 1)$$ and has a horizontal asymptote at $$y = 0.$$

Question 3

Which of the following functions is decreasing for all real values of $$x$$?

Accepted Answer

$$f(x) = \left(\frac{1}{3}\right)^x$$

Question 4

Suppose a population of bacteria triples every hour. If the initial population is 100, which exponential function models the population after $$t$$ hours?

Accepted Answer

$$P(t) = 100 \cdot 3^t$$

Question 5

Which of the following is the correct definition of the number $$e$$?

Accepted Answer

$$e = \lim_{n 	o \infty} \left(1 + \frac{1}{n}ight)^n$$

Question 6

A radioactive substance decays according to $$N(t) = N_0$ \left(\frac{1}{2}\$$right)^{t/5}$$, where t$ is in years. What is the half-life of the substance?

Accepted Answer

5 years

Question 7

Which of the following statements is TRUE about the exponential function $$f(x) = a^x$$ for $$a \u003e 1$$?

Accepted Answer

The function is one-to-one and increasing for all real $$x.$$

Question 8

Solve for $$x$$: $$3^{2x} = 81$.

Accepted Answer

$$x = 2$$

Question 9

A car arrives at a drive-thru at a rate of 0.25 cars per minute. The probability that a car arrives within $$t$$ minutes is $$F(t) = 1 - e$$^{-0.25t}$$. What is the probability that a car arrives within 5 minutes?

Accepted Answer

$$0.713$$

Question 10

Which transformation will shift the graph of $$y = 3$$^{-x}$$ up by 1 unit?

Accepted Answer

$$y = 3$$^{-x}$$ + 1$$

Question 11

Which property of exponential functions explains why the graph of $$f(x) = 2^x$$ never crosses the x-axis?

Accepted Answer

The range of $$f(x) is$$ $$(0, \infty).$$

Question 12

Given the data: $$x = -1, 0, 1, 2, 3$$ and $$y = 81, 27, 9, 3, 1$$, what type of function best models the data?

Accepted Answer

Exponential function

Question 13

If $$f(x) = a^x$$ and $$0 \u003c a \u003c 1$$, which of the following is TRUE?

Accepted Answer

The function is decreasing for all real $$x.$$

Question 14

Which of the following is NOT a property of the exponential function $$f(x) = a^x$$, $$a \u003e 1$$?

Accepted Answer

The function has an x-intercept at $$x = 0.$$

Question 15

Which of the following best describes the end behavior of $$f(x) = 3^x as$$ $$x 	o -\infty$$?

Accepted Answer

$$f(x) 	o 0$$

Question 16

If $$f(x) = 3^x$$ and $$g(x) = 3$$^{-x}$$, how are the graphs of f$ and g$ related?

Accepted Answer

The graph of $$g is $$the reflection of $$f$$ about the $$y-$$axis.

Question 17

Solve for $$x$$: $$e^{2x}$$ = $$e^4.$$

Accepted Answer

$$x = 2$$

Question 18

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

Accepted Answer

The amount of a medication in the bloodstream decreasing by half every hour.

Exponential Functions: Concepts, Properties, and Applications

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