Q: A population of bacteria grows according to the exponential function $$P(t) = 2000 \times 1.05^t$$, where $$t is in $$hours. What is the population after 8 hours?

$$P(8) = 2000 \times 1.05^8$ \approx 2952$$

Q: Which of the following is the correct composition of the functions $$f(x) = 2x + 1$$ and $$g(x) = x^2$ evaluated at x = 3$?

$$f(g(3)) = 2 \times 9 + 1 = 19$$

Q: Which of the following best describes the end behavior of the polynomial function $$f(x) = -2x^4$ + 3x^2$ - 5$$?

As $$x \to \pm \infty$$, $$f(x) \to -\infty$$

Q: A life science researcher models the spread of a virus using the function $$V(t) = 1000 \times e$$^{0.3t}$$, where t$ is days. If the initial population is 1000, what is the population after 10 days?

$$V(10) = 1000 \times e$$^{3}$$ \approx 1000 \times 20.09 = 20090$$

Question 1

Which of the following best describes a function in the context of real-world modeling?

Accepted Answer

A relationship where each input is associated with exactly one output.

Question 2

A pharmaceutical company models the concentration of a drug in the bloodstream over time using the function $$C(t) = 50e$$^{-0.2t}$$, where C$ is concentration in mg/L and t$ is time in hours. What is the concentration after 5 hours?

Accepted Answer

$$C(5) = 50e$$^{-1}$$ \approx 18.4 mg/L$$

Question 3

Which of the following is NOT a key characteristic of polynomial functions?

Accepted Answer

Asymptotes

Question 4

A business models its revenue with the function $$R(x) = 100x - 0.5x^2$, where x$ is the number of units sold. What is the maximum revenue, and at how many units is it achieved?

Accepted Answer

Maximum revenue is $$R(100) = 5000 at$$ $$x = 100$$ units.

Question 5

Which of the following statements about rational functions is correct?

Accepted Answer

Rational functions can have vertical and horizontal asymptotes.

Question 6

Given the cost function $$C(x) = 20x + 500$$ and the revenue function $$R(x) = 50x$$, where $$x is $$the number of items produced and sold, what is the break-even point?

Accepted Answer

Break-even at $$x = 25$$ items

Question 7

Which transformation occurs when the function $$f(x) is $$changed to $$f(x-3)$$?

Accepted Answer

A shift 3 units to the right

Question 8

A population of bacteria grows according to the exponential function $$P(t) = 2000 	imes 1.05^t$$, where $$t is in $$hours. What is the population after 8 hours?

Accepted Answer

$$P(8) = 2000 	imes 1.05^8$ \approx 2952$$

Question 9

Which of the following is the correct composition of the functions $$f(x) = 2x + 1$$ and $$g(x) = x^2$ evaluated at x = 3$?

Accepted Answer

$$f(g(3)) = 2 	imes 9 + 1 = 19$$

Question 10

A social scientist uses a logarithmic function $$S(x) = \ln(x + 1) to $$model survey responses, where $$x is $$the number of years since a policy was implemented. What is $$S(4)$$?

Accepted Answer

$$S(4) = \ln(5) \approx 1.61$$

Question 11

Which of the following best describes the end behavior of the polynomial function $$f(x) = -2x^4$ + 3x^2$ - 5$$?

Accepted Answer

As $$x 	o \pm \infty$$, $$f(x) 	o -\infty$$

Question 12

A company uses regression analysis to predict sales based on advertising spending. Which type of function is most appropriate if the data shows a constant rate of change?

Accepted Answer

Linear function

Question 13

Which of the following is a global property of a function?

Accepted Answer

End behavior

Question 14

A life science researcher models the spread of a virus using the function $$V(t) = 1000 	imes e$$^{0.3t}$$, where t$ is days. If the initial population is 1000, what is the population after 10 days?

Accepted Answer

$$V(10) = 1000 	imes e$$^{3}$$ \approx 1000 	imes 20.09 = 20090$$

Question 15

Which of the following is true about the graph of the rational function $$f(x) = \frac{2x}{x-3}$$?

Accepted Answer

There is a vertical asymptote at $$x = 3$$

Question 16

A manager wants to maximize profit, modeled by $$P(x) = -4x^2$ + 40x - 100$$, where $$x is $$the number of products sold. What is the maximum profit and at how many products?

Accepted Answer

Question 17

Which of the following is the correct solution to the system of equations: $$\begin{cases} 2x + 3y = 12 \ x - y = 2 \end{cases}$$?

Accepted Answer

$$x = 4, y = 2$$

Question 18

Which of the following best describes the concavity of the function $$f(x) = x^3$ - 6x^2$ + 9x$$?

Accepted Answer

Concave up for $$x \u003e 2$$

Functions and Modeling (College Algebra) Syllabus Study Guide

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