Q: A chemist is mixing two solutions: Solution A contains 30% acid and Solution B contains 50% acid. How many liters of each solution should be mixed to obtain 10 liters of a solution that is 40% acid? Let $x$ be the liters of Solution A used. Which equation correctly models this scenario?

$0.3x + 0.5(10 - x) = 0.4 \times 10$

Q: A biologist models the population of a bacteria culture by $P(t) = 1000 + 200t$, where $t$ is time in hours. What does the slope represent in this context?

The population increases by 200 bacteria per hour

Q: Which of the following describes where the function $f(x) = x^3$ is decreasing?

Nowhere; $f(x)$ is always increasing or decreasing depending on the sign of $x$

Question 1

A chemist is mixing two solutions: Solution A contains 30% acid and Solution B contains 50% acid. How many liters of each solution should be mixed to obtain 10 liters of a solution that is 40% acid? Let $x$ be the liters of Solution A used. Which equation correctly models this scenario?

Accepted Answer

$0.3x + 0.5(10 - x) = 0.4 	imes 10$

Question 2

Which of the following equations does NOT represent a function?

Accepted Answer

$x = y^2$

Question 3

Given the function $f(x) = x^3 - 2x$, determine if $f(x)$ is even, odd, or neither.

Accepted Answer

Odd

Question 4

A company’s profit $P$ (in thousands of dollars) after selling $x$ units is modeled by $P(x) = 5x - 200$. What does the $y$-intercept represent in this context?

Accepted Answer

The profit when no units are sold, which is a loss of $200,000

Question 5

Given the function $f(x) = |x - 2|$, what is the domain and range of $f(x)$?

Accepted Answer

Domain: $(-\infty, \infty)$, Range: $[0, \infty)$

Question 6

Which of the following intervals describes where the function $f(x) = -x^2 + 4$ is increasing?

Accepted Answer

$(-\infty, 0)$

Question 7

Given the points $(2, 5)$ and $(6, 13)$, what is the equation of the line passing through these points in slope-intercept form?

Accepted Answer

$y = 2x + 1$

Question 8

A line passes through the point $(3, -2)$ and is perpendicular to the line $y = -\frac{1}{2}x + 4$. What is the equation of the new line?

Accepted Answer

$y + 2 = 2(x - 3)$

Question 9

Evaluate $f(-3)$ for the piecewise function:

$ f(x) = \begin{cases} x^2 + 1 & x < 0 \ 2x - 1 & x \geq 0 \end{cases} $

Accepted Answer

$10$

Question 10

Find the difference quotient for $f(x) = x^2 + 2x$.

Accepted Answer

$2x + h + 2$

Question 11

Which of the following best describes the range of the function $f(x) = \sqrt{x - 3}$?

Accepted Answer

$[0, \infty)$

Question 12

A taxi company charges a flat fee of $\$4$ plus $\$2$ per mile. Which function models the total cost $C$ for $x$ miles?

Accepted Answer

$C(x) = 2x + 4$

Question 13

Given the function $f(x) = x^2 - 4x + 3$, at what $x$-value does the function attain its minimum value?

Accepted Answer

$x = 2$

Question 14

Which of the following is the $x$-intercept of the line $y = 3x - 6$?

Accepted Answer

$(2, 0)$

Question 15

Suppose $f(x)$ is defined as $f(x) = \begin{cases} 2x + 1 & x < 1 \ x^2 & x \geq 1 \end{cases}$. What is $f(1)$?

Accepted Answer

$1$

Question 16

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

Accepted Answer

$\{ x \mid x 
eq 5 \}$

Question 17

A biologist models the population of a bacteria culture by $P(t) = 1000 + 200t$, where $t$ is time in hours. What does the slope represent in this context?

Accepted Answer

The population increases by 200 bacteria per hour

Question 18

Given the function $f(x) = 3x - 7$, evaluate $f(4)$.

Accepted Answer

$5$

Question 19

Which of the following describes where the function $f(x) = x^3$ is decreasing?

Accepted Answer

Nowhere; $f(x)$ is always increasing or decreasing depending on the sign of $x$

Question 20

Which of the following is the correct interval notation for the range of $f(x) = 2x + 5$?

Accepted Answer

$(-\infty, \infty)$

College Algebra Study Guide: Functions and Linear Models

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