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

$x > 2$

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 $f(x) = e^{x-3}$?

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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