Question 1

A company’s profit $$P(x) in $$thousands of dollars is modeled by the function $$P(x) = -2x^2$ + 12x - 8$$, where $$x is $$the number of units produced (in hundreds). What is the maximum profit, and for how many units produced does it occur?

Accepted Answer

The maximum profit is $$10$$ thousand dollars at $$x = 3$$ hundred units.

Question 2

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

Accepted Answer

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

Question 3

Solve the equation $$\sqrt{2x + 3} = x - 1$$ for all real solutions.

Accepted Answer

$$x = 2$$

Question 4

Which of the following is the simplified form of $$\dfrac{2 + 5i}{1 - 2i}$$?

Accepted Answer

$$-1 + 2i$$

Question 5

A radioactive substance decays according to $$A(t) = 100e$$^{-0.05t}$$, where A$ is the amount remaining after t$ years. How many years will it take for the substance to decay to 60$ grams?

Accepted Answer

$$t = \dfrac{\ln(0.6)}{-0.05} \approx 10.22$$ years

Question 6

Which of the following describes the transformation from $$f(x) = x^2$ to g(x) = -2(x$$ - $$3)^2 + 5$?

Accepted Answer

Reflect over the $$x-$$axis, vertically stretch by $$2$$, shift right $$3$$ units, and up $$5$$ units.

Question 7

Given $$f(x) = x^2$ - 4$$ and $$g(x) = \sqrt{x + 2}$$, what is the domain of the composite function $$f \circ g$$?

Accepted Answer

$$x \geq -2$$

Question 8

Solve the inequality $$|2x - 5| \leq 7.$$

Accepted Answer

$$-1 \leq x \leq 6$$

Question 9

Which of the following is the correct simplification of $$\dfrac{3x}{x^2$ - 9} + \dfrac{2}{x + 3}$$?

Accepted Answer

$$\dfrac{3x + 2(x - 3)}{(x + 3)(x - 3)}$$

Question 10

Find all real solutions to the equation $$2^{x+1} = 8$.

Accepted Answer

$$x = 2$$

Question 11

Which of the following is the correct application of the properties of logarithms to simplify $$\log_2$(8x)$$?

Accepted Answer

$$3 + \log_2$(x)$$

Question 12

A rational function is given by $$f(x) = \dfrac{x^2$ - 9}{x^2$ - 4}. $$Which of the following statements is true?

Accepted Answer

The function has vertical asymptotes at $$x = 2$$ and $$x = -2.$$

Question 13

If $$f(x) = x^3$ - 2x is $$increasing on an interval, which of the following intervals is correct?

Accepted Answer

$$(-\infty, -\sqrt{2})$$ and $$(\sqrt{2}, \infty)$$

Question 14

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

Accepted Answer

$$2x + h$$

Question 15

Solve for $$x$$: $$\dfrac{1}{x} + \dfrac{2}{x+1} = 1$$

Accepted Answer

$$x = 1 or$$ $$x = -2$$

Question 16

Which of the following is the correct solution to $$|x^2$ - 4| = 5$$?

Accepted Answer

$$x = 3,\ -3,\ 1,\ -1$$

Question 17

Which of the following is the correct domain for the function $$f(x) = \dfrac{1}{\sqrt{x-2}}$$?

Accepted Answer

$$x \u003e 2$$

Question 18

If $$f(x) = 2x + 1$$ and $$g(x) = x^2$, what is (f$$ \circ g)(3)$$?

Accepted Answer

$$19$$

Question 19

Which of the following is the correct solution to the equation $$x^4$$ - $$5x^2 + 4 = 0$?

Accepted Answer

$$x = 2,\ -2,\ 1,\ -1$$

Precalculus Study Guide: Functions, Equations, and Graphs

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