Q: A car's value $$V in $$dollars after $$t$$ years is modeled by $$V(t) = 20000(0.85)^t. $$What is the value of the car after 3 years?

$$V(3) = 20000 \times 0.85^3$ = 12262.5$$

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

$$(f \circ g)(3) = 2 \times 9 + 1 = 19$$

Q: Which of the following best describes the end behavior of the function $$f(x) = -4x^3$ + 2x$$?

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

Q: A loan of $$5000 is $$compounded annually at a rate of $$6\%. $$What is the value of the loan after $$5$$ years?

$$5000 \times (1.06)^5$ \approx 6691$$

Q: Given the piecewise function $$f(x) = \begin{cases} x+2 & x \u003c 0 \\ 3x & x \geq 0 \end{cases}$$, what is $$f(-3) + f(2)$$?

$$(-3)+2 + 3\times2 = -1 + 6 = 5$$

Question 1

A company models its monthly profit $$P (in $$thousands of dollars) as a function of the number of units sold $$x by$$ $$P(x) = -2x^2$ + 40x - 100. $$What is the maximum monthly profit, and how many units must be sold to achieve it?

Accepted Answer

The maximum profit is $$300$$ thousand dollars, achieved at $$x = 10$$ units.

Question 2

Given the function $$f(x) = 3x - 7$$, what is the inverse function $$f^{-1}(x$$)$$?

Accepted Answer

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

Question 3

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

Accepted Answer

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

Question 4

Which of the following best describes the graph of $$y = -2(x - 3)^2$ + 5$$?

Accepted Answer

A parabola opening downwards with vertex at $$(3, 5)$$

Question 5

Solve the system of equations: $$2x + 3y = 12$$ and $$x - y = 1.$$

Accepted Answer

$$x = 3, y = 2$$

Question 6

Which of the following is the correct domain for the function $$f(x) = \frac{1}{x-5}$$?

Accepted Answer

All real numbers except $$x = 5$$

Question 7

A car's value $$V in $$dollars after $$t$$ years is modeled by $$V(t) = 20000(0.85)^t. $$What is the value of the car after 3 years?

Accepted Answer

$$V(3) = 20000 	imes 0.85^3$ = 12262.5$$

Question 8

Which of the following statements is true about the function $$f(x) = x^2$?

Accepted Answer

It is an even function.

Question 9

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

Accepted Answer

$$(f \circ g)(3) = 2 	imes 9 + 1 = 19$$

Question 10

A radioactive substance decays according to $$A(t) = 100e$$^{-0.05t}$$, where A$ is the amount remaining after t$ years. How much remains after 10 years (rounded to the nearest whole number)?

Accepted Answer

$$A(10) = 100e$$^{-0.5}$$ \approx 61$$

Question 11

Which of the following equations represents a line with slope $$-\frac{3}{2}$$ and $$y-$$intercept $$4$$?

Accepted Answer

$$y = -\frac{3}{2}x + 4$$

Question 12

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

Accepted Answer

$$x = 32$$

Question 13

A quadratic function $$f(x)$$ has its vertex at $$(2, -3)$$ and opens upwards. Which of the following could be its equation?

Accepted Answer

$$f(x) = (x - 2)^2$ - 3$$

Question 14

If $$f(x) = \sqrt{x - 1}$$, what is the domain of $$f$$?

Accepted Answer

$$x \geq 1$$

Question 15

Which of the following best describes the end behavior of the function $$f(x) = -4x^3$ + 2x$$?

Accepted Answer

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

Question 16

A loan of $$5000 is $$compounded annually at a rate of $$6\%. $$What is the value of the loan after $$5$$ years?

Accepted Answer

$$5000 	imes (1.06)^5$ \approx 6691$$

Question 17

Which of the following is the correct solution to the equation $$2^{x} = 20$?

Accepted Answer

$$x = \log_{2}(20)$$

Question 18

Given the piecewise function $$f(x) = \begin{cases} x+2 & x \u003c 0 \ 3x & x \geq 0 \end{cases}$$, what is $$f(-3) + f(2)$$?

Accepted Answer

$$(-3)+2 + 3	imes2 = -1 + 6 = 5$$

Question 19

Which of the following is the correct solution to the quadratic equation $$x^2 - 6x + 5 = 0$?

Accepted Answer

$$x = 1$$ and $$x = 5$$

Question 20

A medication is administered so that its concentration in the bloodstream decreases by $$12\%$$ per hour. If the initial concentration is $$50 mg/L$$, what is the concentration after $$3$$ hours?

Accepted Answer

$$50 	imes (0.88)^3$ \approx 34.1 mg/L$$

College Algebra: Functions and Modeling – Structured Study Notes

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