Q: Which of the following is the correct definition of a logarithmic function?

$$f(x) = \log_b(x)$$,\ where\ b \u003e 0,\ b \neq 1,\ x \u003e 0$

Q: A population of bacteria triples every 4 hours. If the initial population is 200, which function models the population $$P(t)$$ after $$t$$ hours?

$$P(t) = 200 \times 3$$^{t/4}$$

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

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

Q: A loan of $$10,000 is taken out at an annual interest rate of 6\% compounded monthly. What is the amount owed after 3 years? (Use A = P(1$$ + \frac{r}{n}$$)^{nt}$)

$$A = 10,000 \times (1 + \frac{0.06}{12}$$)^{36}$$ \approx 11,972.80$$

Q: A car depreciates in value by 15\% each year. If the car is currently worth $20,000, what will its value be after 4 years?

$$20,000 \times (0.85)^4$ \approx 10,443$$

Q: A savings account earns 5\% annual interest, compounded quarterly. If you deposit $$2,000, how much will you have after 6 years? (Use A = P(1$$ + \frac{r}{n}$$)^{nt}$)

$$A = 2,000 \times (1 + \frac{0.05}{4}$$)^{24}$$ \approx 2,691.59$$

Question 1

A pharmaceutical company models the decay of a drug in the bloodstream using the function $$C(t) = C_0$ e$$^{-0.2t}$$, where C_0$ is the initial concentration and t$ is time in hours. If the initial concentration is 100 mg/L, what is the concentration after 5 hours?

Accepted Answer

$$C(5) = 100 e$$^{-1}$$ \approx 36.79 mg/L$$

Question 2

Which of the following is the correct definition of a logarithmic function?

Accepted Answer

$$f(x) = \log_b(x)$$,\ where\ b \u003e 0,\ b 
eq 1,\ x \u003e 0$

Question 3

A population of bacteria triples every 4 hours. If the initial population is 200, which function models the population $$P(t)$$ after $$t$$ hours?

Accepted Answer

$$P(t) = 200 	imes 3$$^{t/4}$$

Question 4

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

Accepted Answer

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

Question 5

A loan of $$10,000 is taken out at an annual interest rate of 6\% compounded monthly. What is the amount owed after 3 years? (Use A = P(1$$ + \frac{r}{n}$$)^{nt}$)

Accepted Answer

$$A = 10,000 	imes (1 + \frac{0.06}{12}$$)^{36}$$ \approx 11,972.80$$

Question 6

Which of the following equations is equivalent to $$\log_2$(x) = 5$$?

Accepted Answer

$$x = 2^5$

Question 7

A company’s revenue $$R(x) in $$thousands of dollars is modeled by $$R(x) = -2x^2$ + 12x + 20$$, where $$x is $$the number of units sold in thousands. What is the maximum revenue, and for how many units sold does it occur?

Accepted Answer

Maximum revenue is $$R(3) = 38$$ thousand dollars, at $$x = 3$$ thousand units

Question 8

Which of the following is the correct solution to the equation $$2^{x+1} = 16$?

Accepted Answer

$$x = 3$$

Question 9

A chemical reaction follows the law $$A = A_0$ e$$^{-kt}$$. If A_0$ = 50$$ grams and $$A = 25$$ grams after 10 minutes, what is the value of $$k$$?

Accepted Answer

$$k = \dfrac{\ln(2)}{10}$$

Question 10

Which of the following is the correct inverse of the function $$f(x) = 5x - 7$$?

Accepted Answer

$$f^{-1}(x$$) = \dfrac{x+7}{5}$$

Question 11

A car depreciates in value by 15\% each year. If the car is currently worth $20,000, what will its value be after 4 years?

Accepted Answer

$$20,000 	imes (0.85)^4$ \approx 10,443$$

Question 12

Which of the following is the correct solution to the equation $$\log_3$(x) + \log_3$(x-2) = 2$$?

Accepted Answer

$$x = 3$$

Question 13

A rational function has a vertical asymptote at $$x = 2$$ and a horizontal asymptote at $$y = 0. $$Which of the following could be its equation?

Accepted Answer

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

Question 14

If $$f(x) = x^2$ - 4x + 3$$, what is the vertex of the parabola?

Accepted Answer

Vertex at $$(2, -1)$$

Question 15

A savings account earns 5\% annual interest, compounded quarterly. If you deposit $$2,000, how much will you have after 6 years? (Use A = P(1$$ + \frac{r}{n}$$)^{nt}$)

Accepted Answer

$$A = 2,000 	imes (1 + \frac{0.05}{4}$$)^{24}$$ \approx 2,691.59$$

Question 16

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 17

Which of the following is the correct factorization of $$x^3 - 8$?

Accepted Answer

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

Question 18

A radioactive substance decays according to $$N(t) = N_0$ e$$^{-0.05t}$$. If the initial amount is 500 grams, how long will it take for the substance to decay to 100 grams?

Accepted Answer

$$t = \dfrac{\ln(0.2)}{-0.05} \approx 32.19$$ years

Question 19

Which of the following is the correct solution to the equation $$\dfrac{3}{x} + 2 = 5$$?

Accepted Answer

$$x = 1$$

College Algebra Syllabus and Course Structure: Key Topics and Policies

Guida di studio - Domande di pratica

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