Q: Which of the following is the correct domain of the function $$f(x) = \dfrac{1}{x^2$ - 9}$$?

$$x \in \mathbb{R}, x \neq 3, x \neq -3$$

Question 1

A chemist is preparing a solution by mixing two chemicals. The total volume, $$V$$, in liters, is modeled by the equation $$V = 2x + 5$$, where $$x is $$the number of liters of chemical A added. If the chemist wants a total volume of 21 liters, how many liters of chemical A should be used?

Accepted Answer

$$x = 10$$

Question 2

Solve the equation $$\dfrac{3}{x} + 2 = 5$$ for $$x.$$

Accepted Answer

Question 3

A ball is thrown upward and its height in meters after $$t$$ seconds is given by $$h(t) = -4.9t^2$ + 20t + 1.5. $$After how many seconds does the ball reach its maximum height?

Accepted Answer

$$t = 2.04$$

Question 4

Which of the following is the correct solution set for the inequality $$|2x - 3| \u003c 5$$?

Accepted Answer

$$-1 \u003c x \u003c 4$$

Question 5

Given the system of equations:

$$\begin{cases} 2x + 3y = 7 \ x - y = 1 \end{cases}$$

What is the solution $$(x, y)$$?

Accepted Answer

$$(2, 1)$$

Question 6

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

Accepted Answer

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

Question 7

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

Accepted Answer

$$19$$

Question 8

Which of the following functions is the inverse of $$f(x) = 3x - 4$$?

Accepted Answer

$$f^{-1}(x$$) = \dfrac{x + 4}{3}$$

Question 9

A company models its profit with the quadratic function $$P(x) = -2x^2$ + 8x + 10$$, where $$x is $$the number of units sold (in thousands). What is the maximum profit, and for how many units sold does it occur?

Accepted Answer

Maximum profit is $$18 at$$ $$x = 2$$

Question 10

Which of the following is a zero of the polynomial $$f(x) = x^3$ - 4x^2$ + x + 6$$?

Accepted Answer

$$x = -1$$

Question 11

Divide $$2x^3$$ - $$3x^2 + 4x - 5$ by x - 2$. What is the remainder?

Accepted Answer

$$-1$$

Question 12

Solve the inequality $$\dfrac{x + 2}{x - 1} \u003e 0.$$

Accepted Answer

$$x  1$$

Question 13

Express $$\dfrac{5x + 3}{(x + 1)(x - 2)} as a $$sum of partial fractions.

Accepted Answer

$$\dfrac{8}{x + 1} + \dfrac{-3}{x - 2}$$

Question 14

A population of bacteria doubles every 3 hours. If the initial population is 500, which exponential function models the population after $$t$$ hours?

Accepted Answer

$$P(t) = 500 \cdot 2$$^{t/3}$$

Question 15

Which of the following is equivalent to $$\log_2$ 32$$?

Accepted Answer

$$5$$

Question 16

Simplify $$\log_3$ 81 - \log_3$ 9.$$

Accepted Answer

$$1$$

Question 17

Solve for $$x$$: $$2^{x + 1} = 16$.

Accepted Answer

$$x = 3$$

Question 18

A radioactive substance decays according to $$N(t) = N_0$ e$$^{-0.05t}$$, where t$ is in years. If the initial amount is 100 grams, how much remains after 20 years (rounded to the nearest gram)?

Accepted Answer

$$37$$ grams

Question 19

Which of the following is the correct domain of the function $$f(x) = \dfrac{1}{x^2$ - 9}$$?

Accepted Answer

$$x \in \mathbb{R}, x 
eq 3, x 
eq -3$$

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