Q: A $$1000 investment grows continuously at an annual rate of 5$$\%$$. What is its value after 10 years? (Use A$$ = $$Pe^{rt}$)

$$A = 1000 \cdot e$$^{0.05 \times 10}$$

Question 1

A rational function is given by $$f(x) = \frac{x^2$ - 9}{x^2$ - 4x + 3}. $$Which of the following statements about its graph is correct?

Accepted Answer

It has a hole at $$x = 3$$ and a vertical asymptote at $$x = 1.$$

Question 2

Which of the following is the horizontal asymptote of the rational function $$f(x) = \frac{2x^2$ + 5}{x^2$ - 3x + 4}$$?

Accepted Answer

$$y = 2$$

Question 3

Given the exponential function $$f(x) = 3 \cdot 2$$^{x-1}$$, what is the y$-intercept of its graph?

Accepted Answer

$$1.5$$

Question 4

A population of bacteria triples every 4 hours. If the initial population is 200, which equation models the population $$P$$ after $$t$$ hours?

Accepted Answer

$$P = 200 \cdot 3$$^{t/4}$$

Question 5

Which of the following is equivalent to $$\log_2$ 8$$?

Accepted Answer

$$3$$

Question 6

What is the domain of the function $$f(x) = \log_{5}(x-2)$$?

Accepted Answer

$$x \u003e 2$$

Question 7

Which of the following correctly applies the properties of logarithms to simplify $$\log_3$(9x)$$?

Accepted Answer

$$\log_3$ 9 + \log_3$ x$$

Question 8

Solve for $$x$$: $$2^{x+1} = 16$.

Accepted Answer

$$x = 3$$

Question 9

Which of the following is the correct application of the change of base formula for $$\log_7$ 20$$?

Accepted Answer

$$\frac{\log_{10} 20}{\log_{10} 7}$$

Question 10

A $$1000 investment grows continuously at an annual rate of 5$$\%$$. What is its value after 10 years? (Use A$$ = $$Pe^{rt}$)

Accepted Answer

$$A = 1000 \cdot e$$^{0.05 	imes 10}$$

Question 11

Which transformation shifts the graph of $$y = 2^x$$ two units to the right?

Accepted Answer

$$y = 2$$^{x-2}$$

Question 12

Which of the following is the correct domain for the function $$f(x) = \frac{1}{x^2$ - 4}$$?

Accepted Answer

$$x 
eq 2$$ and $$x 
eq -2$$

Question 13

If $$\log_b 2 = 0.356$$ and $$\log_b 5 = 0.699$$, what is $$\log_b 20$$?

Accepted Answer

$$1.055$$

Question 14

Which of the following is the correct inverse of the function $$f(x) = e$$^{x-3}$$?

Accepted Answer

Question 15

Which of the following best describes the end behavior of the rational function $$f(x) = \frac{3x^2$ + 1}{x^2$ - 2x + 4} as$$ $$x 	o \infty$$?

Accepted Answer

$$f(x) 	o 3$$

Question 16

Which of the following is the correct solution to the equation $$\log_4$(x) = 2$$?

Accepted Answer

$$x = 16$$

Question 17

A radioactive substance decays according to $$A = A_0$ e$$^{-0.03t}$$, where t$ is in years. If the initial amount is 500 grams, how much remains after 10 years?

Accepted Answer

$$500 e$$^{-0.3}$$ grams

Question 18

Which of the following is the correct expanded form of $$\ln\left(\frac{e^3$ x^2$}{y}\right)$$?

Accepted Answer

$$3 + 2\ln x - \ln y$$

Question 19

Which of the following is the correct graph for $$y = \log_2$(x-1)$$?

Accepted Answer

It is the graph of $$y = \log_2$ x$$ shifted one unit to the right.

Graphs of Rational, Exponential, and Logarithmic Functions: Key Concepts and Applications

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