Q: Which of the following best describes the end behavior of the polynomial $$f(x) = -2x^5$ + 3x^2$ - 7$$?

As $$x \to \infty$$, $$f(x) \to -\infty$$; as $$x \to -\infty$$, $$f(x) \to \infty$$

Question 1

A pharmaceutical company is testing a new drug and finds that the concentration of the drug in the bloodstream, $$C(t)$$, in mg/L, $$t$$ hours after administration, can be modeled by the function $$C(t) = 8e$$^{-0.5t}$$. What is the concentration after 4 hours?

Accepted Answer

$$C(4) = 8e$$^{-2}$$ \approx 1.08 mg/L$$

Question 2

Which of the following is the correct equation of a line passing through the point $$(2, -3)$$ and perpendicular to the line $$y = 4x + 1$$?

Accepted Answer

$$y + 3 = -\frac{1}{4}(x - 2)$$

Question 3

A biologist models the population of a bacteria culture by $$P(t) = 500 \cdot 2$$^{t}$$, where t$ is measured in hours. How long will it take for the population to reach 4000?

Accepted Answer

$$t = 4$$ hours

Question 4

Which of the following is the correct factorization of the quadratic expression $$6x^2 - 11x - 10$?

Accepted Answer

$$(3x + 2)(2x - 5)$$

Question 5

A company’s profit $$P(x) in $$thousands of dollars is modeled by the quadratic function $$P(x) = -2x^2$ + 12x - 7$$, where $$x is $$the number of units sold in thousands. What is the maximum profit, and how many units must be sold to achieve it?

Accepted Answer

Maximum profit is $$25$$ thousand dollars at $$x = 3$$ thousand units

Question 6

Which of the following is the correct solution set for the inequality $$|2x - 5| \u003c 7$$?

Accepted Answer

$$-1 \u003c x \u003c 6$$

Question 7

Given the function $$f(x) = \frac{2x + 3}{x - 4}$$, what is the domain of $$f(x)$$?

Accepted Answer

$$x 
eq 4$$

Question 8

A projectile is launched from the ground and its height $$h in $$meters after $$t$$ seconds is given by $$h(t) = -4.9t^2$ + 19.6t. At $$what time does the projectile reach its maximum height?

Accepted Answer

$$t = 2$$ seconds

Question 9

Which of the following is the inverse function of $$f(x) = 3x - 7$$?

Accepted Answer

$$f^{-1}(x$$) = \frac{x + 7}{3}$$

Question 10

A chemical reaction’s rate $$R (in $$mol/L·s) is inversely proportional to the square of the concentration $$C (in $$mol/L). If $$R = 0.5$$ when $$C = 2$$, what is $$R$$ when $$C = 4$$?

Accepted Answer

$$R = 0.125$$

Question 11

Which of the following is the correct solution to the system of equations: $$\begin{cases} 2x + 3y = 7 \ x - y = 1 \end{cases}$$?

Accepted Answer

$$x = 2, y = 1$$

Question 12

Which of the following best describes the end behavior of the polynomial $$f(x) = -2x^5$ + 3x^2$ - 7$$?

Accepted Answer

As $$x 	o \infty$$, $$f(x) 	o -\infty$$; as $$x 	o -\infty$$, $$f(x) 	o \infty$$

Question 13

A radioactive substance decays according to the formula $$A(t) = A_0$ e$$^{-kt}$$. If the half-life is 10 years, what is the value of k$?

Accepted Answer

$$k = \frac{\ln 2}{10}$$

Question 14

Which of the following is the correct simplified form of $$\frac{3x^2$ - 12}{6x}$$ for $$x 
eq 0$$?

Accepted Answer

$$\frac{x - 2}{2x}$$

Question 15

A quadratic function $$f(x)$$ has its vertex at $$(1, -4)$$ and passes through the point $$(3, 0). $$Which of the following is its equation?

Accepted Answer

$$f(x) = (x - 1)^2$ - 4$$, but $$a = 1$$ does not satisfy $$(3, 0)$$; $$a = 1 is $$incorrect.

Let’s solve: $$0 = a(3 - 1)^2$ - 4 \implies 0 = 4a - 4 \implies a = 1$$
So $$f(x) = (x - 1)^2$ - 4$$

Question 16

Which of the following is the correct solution to the equation $$\log_2$(x - 1) = 3$$?

Accepted Answer

Question 17

A hospital is monitoring the temperature of a patient. The temperature $$T in $$degrees Celsius at time $$t$$ hours is modeled by $$T(t) = 37 + 2\sin\left(\frac{\pi}{12}t\right). $$What is the maximum temperature reached by the patient?

Accepted Answer

$$39$$ degrees Celsius

Question 18

Which of the following is the correct solution to the equation $$2^{x+1} = 32$?

Accepted Answer

$$x = 4$$

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