Q: The function $f(x) = |x|$ can be written as a piecewise function. Which of the following is correct?

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

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

$(-\\infty, 1) \\cup (1, \\infty)$

Question 1

Which of the following represents the interval notation for all real numbers $x$ such that $2 < x \leq 7$?

Accepted Answer

$(2, 7]$

Question 2

Given the function $f(x) = \sqrt{1 - x^2}$, what is the domain of $f(x)$ in interval notation?

Accepted Answer

$[-1, 1]$

Question 3

A box is to be constructed from a $6 \times 8$ piece of cardboard by cutting out squares of side $x$ from each corner and folding up the sides. Which of the following expressions gives the volume $V(x)$ of the box as a function of $x$?

Accepted Answer

$V(x) = x(6 - 2x)(8 - 2x)$

Question 4

For the box construction problem, what is the appropriate domain for $x$ based on physical constraints?

Accepted Answer

$[0, 3]$

Question 5

Which of the following is the correct interval notation for the set $\{x \mid x \geq 5\}$?

Accepted Answer

$[5, \infty)$

Question 6

The function $f(x) = |x|$ can be written as a piecewise function. Which of the following is correct?

Accepted Answer

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

Question 7

Which of the following is the vertex of the quadratic function $f(x) = 2x^2 - 4x + 1$?

Accepted Answer

$(1, -1)$

Question 8

For the quadratic $f(x) = ax^2 + bx + c$, what is the formula for the $x$-coordinate of the vertex?

Accepted Answer

$x = -\frac{b}{2a}$

Question 9

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

Accepted Answer

$(0, \infty)$

Question 10

A piecewise function is defined as $f(x) = \begin{cases} 2x + 2 & x < 1 \ \ -3x + 5 & x \geq 1 \end{cases}$. What is the value of $f(1)$?

Accepted Answer

$2$

Question 11

Which of the following is the correct interval notation for the set $\{x \mid x < 0 \}$?

Accepted Answer

$(-\infty, 0)$

Question 12

If $f(x) = x^2 + 4x + 5$, what is the minimum value of $f(x)$?

Accepted Answer

$1$

Question 13

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

Accepted Answer

$(-\infty, 1) \cup (1, \infty)$

Question 14

A polynomial of degree $n > 2$ can have which of the following features that a quadratic cannot?

Accepted Answer

Multiple peaks and valleys (local maxima and minima)

Question 15

Which of the following statements about the roots of the quadratic $f(x) = ax^2 + bx + c$ is true?

Accepted Answer

There are two distinct real roots if $b^2 - 4ac > 0$

Question 16

Suppose a function $f(x)$ is defined as $f(x) = \begin{cases} 3x - 1 & x < 2 \ \ x^2 & x \geq 2 \end{cases}$. What is the value of $f(2)$?

Accepted Answer

$4$

Question 17

Which of the following is the correct interval notation for the set of all real numbers?

Accepted Answer

$(-\infty, \infty)$

Functions, Domains, and Graphs: Foundations for Business Calculus

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