Question 1

Given the function $$f(x) = x^3$ - 6x^2$ + 9x + 1$$, which of the following intervals is where the function is decreasing?

Accepted Answer

$$(1, 3)$$

Question 2

Suppose a ball is thrown vertically upward from the top of a building 800 feet tall with an initial velocity of 80 feet per second. The distance $$d(t) in $$feet of the ball from the ground after $$t$$ seconds is given by $$d(t) = -16t^2$ + 80t + 800. $$What is the maximum height of the ball?

Accepted Answer

$$1000$$ feet

Question 3

Find the domain of the function $$f(x) = \dfrac{2x}{x^2$ - 9}.$$

Accepted Answer

$$x 
eq -3,\ 3$$

Question 4

A radioactive substance decays according to the model $$A(t) = Ae$$^{-0.07t}$$, where A$ is the initial amount present and t$ is time in years. After how many years will 80\% of the original amount remain?

Accepted Answer

$$t = 3.19$$

Question 5

Which of the following is the correct sum of $$\log(x) + \log(x-3)$$ expressed as a single logarithm?

Accepted Answer

$$\log(x(x-3))$$

Question 6

Given $$f(x) = x^4$ - 9x$$ and $$g(x) = 2x + 3$$, what is $$f(g(x))$$?

Accepted Answer

$$(2x+3)^4 - 9(2x+3$$)$$

Question 7

A company’s profit is modeled by $$P(x) = -0.04x^2$ + 90x - 300$$, where $$x is $$the number of items sold. What is the maximum profit, and how many items must be sold to reach it?

Accepted Answer

Maximum profit is $$10125 at$$ $$x = 1125$$

Question 8

Which of the following functions is even?

Accepted Answer

$$f(x) = x^4$ + 5x^2$

Question 9

Solve for $$x$$: $$6e^{x+9} = 30$

Accepted Answer

$$x = \ln(5) - 9$$

Question 10

Suppose a polynomial function of degree 4 with rational coefficients has zeros $$-4$$, $$1$$, and $$2i. $$What is the polynomial written as a product of linear factors?

Accepted Answer

$$(x+4)(x-1)(x-2i)(x+2i)$$

Question 11

A large group of students is asked to memorize a list of 60 Italian words. The number of words retained is modeled by $$W(t) = 60e$$^{-0.2t}$$, where t$ is the number of hours after memorization. What is the horizontal asymptote for the graph of this function?

Accepted Answer

$$y = 0$$

Question 12

Given the piecewise-defined function $$f(x) = \begin{cases} 2x+1 & x \u003c -2 \ x^2$ & x \geq -2 \end{cases}$$, what is $$f(-3)$$?

Accepted Answer

$$f(-3) = -5$$

Question 13

Convert the equation $$4^x = 2.8 to a $$logarithmic equation.

Accepted Answer

$$x = \log_4$(2.8)$$

Question 14

Which of the following is the correct expression for the difference quotient $$\dfrac{f(x+h) - f(x)}{h}$$ for $$f(x) = x^2$?

Accepted Answer

$$2x + h$$

Question 15

A certain company has fixed costs of $$1500 with a variable cost per unit of 22. $$The commodity is sold for $$30$$ per unit. Which equation represents the break-even point?

Accepted Answer

$$1500 + 22x = 30x$$

Question 16

Which of the following is the correct solution to $$\log(x+1) - \log(x-3) = \log(2)$$?

Accepted Answer

$$x = 5$$

Question 17

Given the function $$f(x) = \dfrac{3x+1}{x-2}$$, what is the equation of the vertical asymptote?

Accepted Answer

$$x = 2$$

Question 18

Which of the following is the correct inverse of $$f(x) = 3x - 2$$?

Accepted Answer

$$f^{-1}(x$$) = \dfrac{x+2}{3}$$

Question 19

If the graph of $$f(x) is $$reflected over the y-axis and shifted 2 units down, what is the resulting function?

Accepted Answer

$$f(-x) - 2$$

College Algebra Final Exam Review: Comprehensive Study Notes

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