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

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

Q: A car's value depreciates exponentially according to $$V(t) = 25000(0.85)^t$$, where $$t is $$the number of years after purchase. What is the value of the car after 3 years?

$$V(3) = 25000 \times (0.85)^3$

Question 1

A company models its profit $$P (in $$thousands of dollars) as a function of time $$t (in $$years) by the equation $$P(t) = 5t^2$ - 20t + 30. $$What is the minimum profit, and after how many years does it occur?

Accepted Answer

The minimum profit is $$10$$ thousand dollars, occurring at $$t = 2$$ years.

Question 2

Which of the following 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 3

A biologist models the population of a bacteria culture by $$P(t) = 200e$$^{0.3t}$$, where t$ is in hours. How long will it take for the population to triple?

Accepted Answer

$$t = \dfrac{\ln 3}{0.3}$$ hours

Question 4

Which of the following is the correct inverse of the function $$f(x) = 2x - 5$$?

Accepted Answer

$$f^{-1}(x$$) = \dfrac{x + 5}{2}$$

Question 5

A rational function is given by $$f(x) = \dfrac{3x^2$ - 2x + 1}{x^2$ - 4}. $$Which of the following describes its vertical asymptotes?

Accepted Answer

Vertical asymptotes at $$x = 2$$ and $$x = -2$$

Question 6

A ball is thrown upward from the top of a 50-meter building with an initial velocity of 20 m/s. Its height $$h (in $$meters) after $$t$$ seconds is given by $$h(t) = -4.9t^2$ + 20t + 50. $$After how many seconds does the ball reach its maximum height?

Accepted Answer

$$t = 2.04$$ seconds

Question 7

Which of the following equations represents a circle with center at $$(3, -2)$$ and radius $$5$$?

Accepted Answer

$$(x - 3)^2$ + (y + 2)^2$ = 25$$

Question 8

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

Accepted Answer

$$x \geq 4$$

Question 9

A radioactive substance decays according to $$A(t) = 100e$$^{-0.05t}$$, where A$ is the amount remaining after t$ years. What is the half-life of the substance?

Accepted Answer

$$t = \dfrac{\ln 2}{0.05}$$ years

Question 10

Which of the following best describes the graph of $$y = |x - 3| + 2$$?

Accepted Answer

A V-shaped graph with vertex at $$(3, 2)$$

Question 11

If $$f(x) = 2x + 1$$ and $$g(x) = x^2$, what is (f$$ \circ g)(x)$$?

Accepted Answer

$$2x^2 + 1$

Question 12

A car's value depreciates exponentially according to $$V(t) = 25000(0.85)^t$$, where $$t is $$the number of years after purchase. What is the value of the car after 3 years?

Accepted Answer

$$V(3) = 25000 	imes (0.85)^3$

Question 13

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

Accepted Answer

$$x \u003c 6$$

Question 14

Given $$f(x) = \dfrac{1}{x-1}$$, which of the following statements is true?

Accepted Answer

The function has a vertical asymptote at $$x = 1$$

Question 15

A quadratic equation $$ax^2 + bx + c = 0$ has real and equal roots if:

Accepted Answer

$$b^2 - 4ac = 0$

Question 16

Which of the following is the solution to the equation $$\log_2$(x) = 5$$?

Accepted Answer

$$x = 32$$

Question 17

A parabola has its vertex at $$(2, -3)$$ and opens upward. Which of the following could be its equation?

Accepted Answer

$$y = a(x - 2)^2$ - 3$$, where $$a \u003e 0$$

Question 18

Which of the following is the correct equation for the line passing through the points $$(1, 2)$$ and $$(3, 6)$$?

Accepted Answer

$$y = 2x$$

Question 19

If $$f(x) = x^2$ - 4x + 7$$, what is the minimum value of $$f(x)$$?

Accepted Answer

$$3$$

Question 20

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

Accepted Answer

$$x = 2, y = 5$$

College Algebra Syllabus and Core Concepts Study Guide

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