Question 1

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

Accepted Answer

The function is increasing if $$a \u003e 1$$ and decreasing if $$0 \u003c a \u003c 1.$$

Question 2

What is the domain and range of the exponential function $$f(x) = a^x$$ where $$a \u003e 0$$ and $$a 
eq 1$$?

Accepted Answer

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

Question 3

If $$a^x = a^y$$ for $$a \u003e 0$$ and $$a 
eq 1$$, what is the relationship between $$x$$ and $$y$$?

Accepted Answer

$$x = y$$

Question 4

Solve for $$x$$: $$9^x = 27.$$

Accepted Answer

$$x = \frac{3}{2}$$

Question 5

Which property of exponents is illustrated by $$a^x \cdot a^y = a$$^{x+y}$$?

Accepted Answer

Product of powers property

Question 6

If a bacterial culture starts with 500 bacteria and increases by 30% every hour, what is the population after 3 hours?

Accepted Answer

$$500 	imes 1.3^3$

Question 7

Which of the following is the natural exponential function?

Accepted Answer

$$f(x) = e^x$$

Question 8

Solve for $$x$$: $$32^{2x+1}$$ = $$128^{x+3}.$$

Accepted Answer

$$x = 4$$

Question 9

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

Accepted Answer

$$a^x + a^y = a$$^{x+y}$$

Question 10

If $$f(x) = e^x$$, which of the following is TRUE?

Accepted Answer

$$f(x) is $$increasing for all $$x.$$

Question 11

Solve for $$x$$: $$e^{x-1} = (e$$^$$x)^{1/3}.$$

Accepted Answer

$$x = \frac{3}{2}$$

Question 12

Which of the following is equivalent to $$a^{-x}$$?

Accepted Answer

$$\frac{1}{a^x}$$

Question 13

If $$f(x) = a^x$$ and $$a = 1$$, what is the graph of $$f(x)$$?

Accepted Answer

A horizontal line at $$y = 1$$

Question 14

Which point is always on the graph of $$f(x) = a^x$$ for $$a \u003e 0$$ and $$a 
eq 1$$?

Accepted Answer

$$(0, 1)$$

Question 15

If $$f(x) = a^x$$ and $$a \u003c 1$$, which best describes the graph?

Accepted Answer

It is decreasing as $$x$$ increases.

Question 16

Solve for $$x$$: $$7^{x+1} = 1$.

Accepted Answer

$$x = -1$$

Question 17

Which of the following is a correct application of exponential growth in a real-world scenario?

Accepted Answer

A population doubles every fixed period, modeled by $$P(t) = P_0$ \cdot 2^t$$

Question 18

Which of the following is NOT a property of the natural exponential function $$f(x) = e^x$$?

Accepted Answer

$$e^x + e^y = e$$^{x+y}$$

Question 19

If $$f(x) = a^x$$ and $$a \u003e 1$$, which of the following best describes the end behavior as $$x 	o \infty$$?

Accepted Answer

$$f(x) 	o \infty$$

Question 20

Which of the following equations models a population that grows by 30% every hour, starting with 500 bacteria?

Accepted Answer

$$P(t) = 500 \cdot 1.3^t$$

Exponential Functions in Calculus: Properties, Laws, and Applications

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