Q: A company’s profit $$P is a $$function of the number of units sold $$x$$, given by $$P(x) = 4x - 20. $$What is the profit if 10 units are sold?

$$P(10) = 4 \times 10 - 20 = 40 - 20 = 20$$

Q: A business’s revenue $$R is $$modeled by $$R(x) = 100x - x^2$, where x$ is the number of units sold. What is the maximum revenue, and for how many units is it achieved?

Maximum revenue is $$R(50) = 100 \times 50 - 50^2$ = 5000 - 2500 = 2500$$, achieved at $$x = 50$$

Question 1

A pharmaceutical company needs to ensure that the concentration of a drug in a solution stays between 2 mg/mL and 5 mg/mL for safety. Which inequality correctly models the acceptable concentration $$c$$?

Accepted Answer

$$2 \leq c \leq 5$$

Question 2

A delivery drone travels from point $$A(2, 3) to $$point $$B(8, 15). $$What is the distance between $$A$$ and $$B$$?

Accepted Answer

$$\sqrt{(8-2)^2$ + (15-3)^2$} = \sqrt{36 + 144} = \sqrt{180} \approx 13.42$$

Question 3

The midpoint between two cities located at $$(4, -2)$$ and $$(10, 8) is $$used to build a rest stop. What are the coordinates of the midpoint?

Accepted Answer

$$\left( \frac{4+10}{2}, \frac{-2+8}{2} \right) = (7, 3)$$

Question 4

A circular fountain in a park has its center at $$(3, 4)$$ and a radius of $$5$$ meters. Which equation represents the fountain's boundary?

Accepted Answer

$$(x-3)^2$$ + $$(y-4)^2 = 25$

Question 5

A company’s profit $$P is a $$function of the number of units sold $$x$$, given by $$P(x) = 4x - 20. $$What is the profit if 10 units are sold?

Accepted Answer

$$P(10) = 4 	imes 10 - 20 = 40 - 20 = 20$$

Question 6

Which of the following is the parent function for quadratic equations?

Accepted Answer

$$f(x) = x^2$

Question 7

A piecewise function models a taxi fare: $$f(x) = \begin{cases} 3 & x \leq 2 \ 3 + 2(x-2) & x \u003e 2 \end{cases}$$, where $$x is $$the number of miles. What is the fare for a 5-mile trip?

Accepted Answer

$$f(5) = 3 + 2(5-2) = 3 + 6 = 9$$

Question 8

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

Accepted Answer

$$(g \circ h)(2) = g(h(2)) = g(3) = 3^2$ = 9$$

Question 9

A linear equation models the temperature $$T in a $$room as a function of time $$t in $$hours: $$T(t) = 20 + 3t. $$What is the slope of the line, and what does it represent?

Accepted Answer

Slope is $$3$$, representing the rate of temperature increase per hour.

Question 10

Which transformation occurs when the function $$f(x) = x^2$ is changed to f(x$$) = $$(x-4)^2 + 2$?

Accepted Answer

Shift right by 4 units and up by 2 units.

Question 11

Solve the absolute value equation: $$|2x - 5| = 7$$

Accepted Answer

$$x = 6 or$$ $$x = -1$$

Question 12

A chemical reaction requires the temperature $$T to $$satisfy $$|T - 100| \u003c 5. $$What is the acceptable range for $$T$$?

Accepted Answer

$$95 \u003c T \u003c 105$$

Question 13

Solve the quadratic equation: $$x^2 - 6x + 8 = 0$

Accepted Answer

$$x = 2 or$$ $$x = 4$$

Question 14

Solve the rational equation: $$\frac{2}{x} = 4$$

Accepted Answer

$$x = 0.5$$

Question 15

Solve the radical equation: $$\sqrt{x + 9} = 7$$

Accepted Answer

$$x = 40$$

Question 16

Which of the following equations is in quadratic form?

Accepted Answer

$$x^4$$ - $$5x^2 + 6 = 0$

Question 17

A business’s revenue $$R is $$modeled by $$R(x) = 100x - x^2$, where x$ is the number of units sold. What is the maximum revenue, and for how many units is it achieved?

Accepted Answer

Maximum revenue is $$R(50) = 100 	imes 50 - 50^2$ = 5000 - 2500 = 2500$$, achieved at $$x = 50$$

Question 18

A line passes through $$(1, 2)$$ and $$(4, 8). $$What is its equation in slope-intercept form?

Accepted Answer

$$y = 2x$$

Question 19

A hospital uses the function $$f(x) = |x - 98.6| to $$monitor patient temperatures. If $$f(x) \u003e 2$$, what does this indicate?

Accepted Answer

The patient’s temperature deviates by more than 2 degrees from normal.

College Algebra Study Guide: Inequalities, Functions, Graphs, and Equations

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