Skip to main content
Contents Index Dark Mode Prev Up Next \(\newcommand{\N}{\mathbb N} \newcommand{\Z}{\mathbb Z} \newcommand{\Q}{\mathbb Q} \newcommand{\R}{\mathbb R}
\newcommand{\lt}{<}
\newcommand{\gt}{>}
\newcommand{\amp}{&}
\definecolor{fillinmathshade}{gray}{0.9}
\newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}}
\)
Section 5.2 Quizzes on Functions
This section contains interactive quizzes and exercises for students to practice the concepts covered in the book. Each quiz includes multiple choice questions, open-ended questions, and exercises that require students to show their work. The quizzes are designed to reinforce understanding and provide immediate feedback on key topics such as solving inequalities, graphing functions, identifying asymptotes, and analyzing points of discontinuity. Students are encouraged to attempt each question and review the explanations provided for the correct answers to deepen their comprehension of the material.
Subsection 5.2.1 Determining if an Equation is a Function
Checkpoint 5.2.1 .
Which of the following describes the fundamental requirement for a relation to be a function?
Every x-value must be positive.
Every input must have exactly one output.
The graph must be a straight line.
The equation must contain a square root.
Checkpoint 5.2.2 .
Does the equation
\(x^2 + y^2 = 25\) represent
\(y\) as a function of
\(x\text{?}\)
Yes, because it is a circle.
Yes, because every x has a y.
No, because it fails the Vertical Line Test.
No, because it is not a polynomial.
Checkpoint 5.2.3 .
Determine if the relation
\(\{(1, 2), (2, 3), (3, 2), (4, 5)\}\) is a function.
No, because the output ’2’ is repeated.
No, because it is not an equation.
Yes, because it passes the Horizontal Line Test.
Checkpoint 5.2.4 .
Which test is used to determine if a graph represents a function?
Checkpoint 5.2.5 .
Solve for
\(y\) in
\(x = y^2\text{.}\) Is
\(y\) a function of
\(x\text{?}\)
Yes,
\(y = \sqrt{x}\text{.}\)
No, because
\(y = \pm \sqrt{x}\) provides two outputs for one input.
Yes, because it is a parabola.
No, because
\(x\) cannot be negative.
Checkpoint 5.2.6 .
Is the linear equation
\(3x - 4y = 12\) a function?
Yes, all non-vertical lines are functions.
No, because it has an x-intercept.
No, because it is a relation.
Yes, but only if it passes through the origin.
Checkpoint 5.2.7 .
Given
\(f(x) = |x|\text{,}\) is this a function?
No, because it has a sharp corner at (0,0).
No, because positive and negative x values give the same y.
Yes, because each x results in exactly one absolute value.
Yes, because it is a one-to-one function.
Checkpoint 5.2.8 .
Which of the following equations is
NOT a function of
\(x\text{?}\)
Checkpoint 5.2.9 .
In the context of calculus, if a curve fails the Vertical Line Test, we can still analyze it by:
Ignoring the vertical parts.
Restricting the domain or using parametric equations.
Declaring it "undefined" in all cases.
Treating
\(x\) as the dependent variable only.
Checkpoint 5.2.10 .
Does the table below represent a function?
Table 5.2.11.
Yes, because all y values are different.
No, because the input x=2 maps to both y=1 and y=6.
Yes, because it is a finite set.
No, because the x values are not in order.
Subsection 5.2.2 Ploting of Functions
Checkpoint 5.2.12 .
1. Plot the graph of: \( y = 2x \)
Is this a function? Explain why or why not.
Answer .
[Student writes explanation and shows graph here]
Checkpoint 5.2.13 .
2. Plot the graph of: \( y = 2x^2 - 5 \)
Is this a function? Explain why or why not.
Answer .
[Student writes explanation and shows graph here]
Checkpoint 5.2.14 .
3. Plot the graph of: \( x^2 + y^2 = 25 \)
Is this a function? Explain why or why not.
Answer .
[Student writes explanation and shows graph here]
Checkpoint 5.2.15 .
4. Plot the graph of: \( xy = 1 \)
Is this a function? Explain why or why not.
Answer .
[Student writes explanation and shows graph here]
Checkpoint 5.2.16 .
5. Plot the graph of: \( x = 5 \)
Is this a function? Explain why or why not.
Answer .
[Student writes explanation and shows graph here]
Checkpoint 5.2.17 .
1. \( f(x) = \begin{cases} 2x & x \lt 0 \\ x^2 & x \ge 0 \end{cases} \)
Answer .
[Student plots the graph and discusses domain, range, continuity, etc.]
Checkpoint 5.2.18 .
Answer .
[Student plots the graph and discusses domain, range, continuity, etc.]
Checkpoint 5.2.19 .
Answer .
[Student plots the graph and discusses domain, range, continuity, etc.]
Checkpoint 5.2.20 .
Answer .
[Student plots the graph and discusses domain, range, continuity, etc.]
Checkpoint 5.2.21 .
5. \( f(x) = \sqrt[3]{x} \)
Answer .
[Student plots the graph and discusses domain, range, continuity, etc.]
Checkpoint 5.2.22 .
6. \( f(x) = \frac{1}{x} \)
Answer .
[Student plots the graph and discusses domain, range, continuity, etc.]
Checkpoint 5.2.23 .
Answer .
[Student plots the graph and discusses domain, range, continuity, etc.]
Checkpoint 5.2.24 .
Answer .
[Student plots the graph and discusses domain, range, continuity, etc.]
Checkpoint 5.2.25 .
Answer .
[Student plots the graph and discusses domain, range, continuity, etc.]
Checkpoint 5.2.26 .
Answer .
[Student plots the graph and discusses domain, range, continuity, etc.]
Subsection 5.2.3 Greatest Integer Function, \(\lfloor x \rfloor\) and Fractional Part Function, \(\{x\}\)
Checkpoint 5.2.27 .
What is the value of
\(\lfloor 2.7 \rfloor\text{,}\) where
\(\lfloor x \rfloor\) denotes the greatest integer function?
Checkpoint 5.2.28 .
Evaluate the fractional part function
\(\{x\}\) for
\(x = -1.3\text{.}\)
Checkpoint 5.2.29 .
Solve for
\(x\text{:}\) \(\lfloor x \rfloor = 5\text{.}\)
Checkpoint 5.2.30 .
Which of the following is an identity for the fractional part function?
\(\{x\} = x + \lfloor x \rfloor\)
\(\{x\} = x - \lfloor x \rfloor\)
\(\{x\} = \lfloor x \rfloor - x\)
\(\{x\} = \lfloor x + 1 \rfloor\)
Checkpoint 5.2.31 .
What is the value of
\(\lfloor -3.1 \rfloor\text{?}\)
Checkpoint 5.2.32 .
If
\(f(x) = \{x\}\text{,}\) what is the period of the function?
Checkpoint 5.2.33 .
Evaluate
\(\lfloor \pi \rfloor + \{\pi\}\text{.}\)
Checkpoint 5.2.34 .
Solve the equation
\(\{x\} = 0\text{.}\)
\(x\) is any real number.
\(x\) is any irrational number.
Checkpoint 5.2.35 .
What is the range of the function
\(f(x) = \{x\}\text{?}\)
Checkpoint 5.2.36 .
Evaluate
\(\lfloor -\sqrt{2} \rfloor\text{.}\)
Subsection 5.2.4 Even and Odd Functions
Checkpoint 5.2.37 .
If
\(f(x) = x^4 - 3x^2 + 7\text{,}\) the function is:
Checkpoint 5.2.38 .
Which statement describes the symmetry of an odd function
\(g(x)\text{?}\)
It is symmetric about the x-axis.
It is symmetric about the y-axis.
It is symmetric about the origin.
It is symmetric about the line
\(y = x\text{.}\)
Checkpoint 5.2.39 .
Determine the nature of
\(f(x) = \sin(x) + x\text{.}\)
Periodic but neither even nor odd
Checkpoint 5.2.40 .
If
\(h(x)\) is an even function and
\(h(3) = -4\text{,}\) what is
\(h(-3)\text{?}\)
Checkpoint 5.2.41 .
What is the result of multiplying an even function by an odd function?
Checkpoint 5.2.42 .
Which of the following functions is an odd function?
Checkpoint 5.2.43 .
The graph of an even function is symmetric with respect to the:
Checkpoint 5.2.44 .
Determine if
\(f(x) = x \cdot \sin(x)\) is even, odd, or neither.
Checkpoint 5.2.45 .
If
\(f(x)\) is an odd function and
\(f(2) = 5\text{,}\) what is the value of
\(f(-2)\text{?}\)
Checkpoint 5.2.46 .
The product of two odd functions is always:
Subsection 5.2.5 Determing the Domain of Functions
Checkpoint 5.2.47 .
Find the domain of
\(f(x) = \frac{5}{2x - 10}\text{.}\)
Checkpoint 5.2.48 .
What is the domain of
\(g(x) = \sqrt{7 - x}\text{?}\)
Checkpoint 5.2.49 .
Determine the domain of
\(h(x) = \log_{10}(x^2 - 9)\text{.}\)
\((-\infty, -3) \cup (3, \infty)\)
All real numbers except 3 and -3
Checkpoint 5.2.50 .
Find the domain of
\(f(x) = \frac{1}{\sqrt{x + 2}}\text{.}\)
Checkpoint 5.2.51 .
What is the domain of the function
\(f(x) = \frac{x + 1}{x^2 + 1}\text{?}\)
Checkpoint 5.2.52 .
The domain of
\(f(x) = \arcsin(x)\) is:
Checkpoint 5.2.53 .
Find the domain of
\(g(x) = \frac{3}{x^2 - 5x + 6}\text{.}\)
Checkpoint 5.2.54 .
Determine the domain of
\(f(x) = \sqrt[3]{x - 1}\) (cube root).
Checkpoint 5.2.55 .
What is the domain of
\(f(x) = e^{1/x}\text{?}\)
Checkpoint 5.2.56 .
Find the domain of
\(h(x) = \sqrt{x^2 - 4x}\text{.}\)
\((-\infty, 0] \cup [4, \infty)\)
Checkpoint 5.2.57 .
A function
\(f(x)\) is considered ’even’ if it satisfies which of the following conditions for all
\(x\) in its domain?
Checkpoint 5.2.58 .
What is the domain of the function
\(f(x) = \frac{1}{x^2 - 4}\text{?}\)
All real numbers except
\(x = 4\)
All real numbers except
\(x = 2\) and
\(x = -2\)
Checkpoint 5.2.59 .
Find the domain of the function
\(f(x) = \sqrt{x - 5}\text{.}\)
Checkpoint 5.2.60 .
What is the domain of
\(f(x) = \ln(x + 3)\text{?}\)
Checkpoint 5.2.61 .
Which function has a domain of all real numbers?
Subsection 5.2.6 Determing the Range of Functions
Checkpoint 5.2.62 .
What is the range of the function
\(f(x) = x^2 + 3\text{?}\)
Checkpoint 5.2.63 .
Find the range of
\(g(x) = \sin(x) - 5\text{.}\)
Checkpoint 5.2.64 .
Determine the range of the constant function
\(f(x) = 7\text{.}\)
All real numbers except 7
Checkpoint 5.2.65 .
What is the range of
\(h(x) = \sqrt{x - 2}\text{?}\)
Checkpoint 5.2.66 .
Find the range of
\(f(x) = |x| + 1\text{.}\)
Checkpoint 5.2.67 .
What is the range of
\(f(x) = \frac{1}{x}\) for
\(x \neq 0\text{?}\)
\(\{ y \in \mathbb{R} \mid y \neq 0 \}\)
\(\{ y \in \mathbb{R} \mid y > 0 \}\)
Checkpoint 5.2.68 .
Determine the range of
\(f(x) = -x^2 + 4\text{.}\)
Checkpoint 5.2.69 .
What is the range of the exponential function
\(f(x) = e^x\text{?}\)
Checkpoint 5.2.70 .
Find the range of
\(f(x) = 2x + 5\) over the domain
\(\mathbb{R}\text{.}\)
Checkpoint 5.2.71 .
If
\(f(x) = x^2\) and the domain is restricted to
\([1, 3]\text{,}\) what is the range?
Checkpoint 5.2.72 .
What is the range of the natural logarithmic function
\(f(x) = \ln(x)\text{?}\)
Checkpoint 5.2.73 .
Find the range of
\(g(x) = |x - 4| - 2\text{.}\)
Checkpoint 5.2.74 .
What is the range of the fractional part function
\(f(x) = \{x\}\text{?}\)
Checkpoint 5.2.75 .
Determine the range of
\(h(x) = \log_{10}(x^2 + 10)\text{.}\)
Checkpoint 5.2.76 .
Find the range of
\(f(x) = 3 - |x + 1|\text{.}\)
Subsection 5.2.7 Exponentail and Logarithms Functions
Checkpoint 5.2.77 .
What is the value of
\(\log_2(32)\text{?}\)
Checkpoint 5.2.78 .
Which of the following is equivalent to
\(\ln(e^x)\text{?}\)
Checkpoint 5.2.79 .
Expand the expression:
\(\log(ab^2)\)
\(\log(a) \cdot 2\log(b)\)
Checkpoint 5.2.80 .
Solve for
\(x\text{:}\) \(\log_3(x) = 4\)
Checkpoint 5.2.81 .
What is the value of
\(\ln(1)\text{?}\)
Checkpoint 5.2.82 .
Express
\(\log_5(10)\) using the change of base formula with natural logarithms.
\(\frac{\ln(5)}{\ln(10)}\)
\(\frac{\ln(10)}{\ln(5)}\)
Checkpoint 5.2.83 .
Simplify the expression:
\(\ln(e^3) + \ln(e^2)\)
Checkpoint 5.2.84 .
Which of the following is the domain of the function
\(f(x) = \log(x - 3)\text{?}\)
Checkpoint 5.2.85 .
Solve for
\(x\text{:}\) \(e^{2x} = 7\)
Checkpoint 5.2.86 .
What is the result of
\(\log(100)\) if no base is explicitly written?
Subsection 5.2.8 Composite Functions
Checkpoint 5.2.87 .
Given
\(f(x) = 2x + 3\) and
\(g(x) = x^2\text{,}\) \((f \circ g)(x) =\)
Checkpoint 5.2.88 .
Given
\(f(x) = \sqrt{x}\) and
\(g(x) = x - 4\text{,}\) \((f \circ g)(x) =\)
Checkpoint 5.2.89 .
Given
\(f(x) = 3x - 1\) and
\(g(x) = 2x + 5\text{,}\) \((g \circ f)(x) =\)
Checkpoint 5.2.90 .
Given
\(f(x) = x^2 + 1\) and
\(g(x) = \frac{1}{x}\text{,}\) \((f \circ g)(x) =\)
Checkpoint 5.2.91 .
Given
\(f(x) = x - 2\) and
\(g(x) = x^2\text{,}\) \((g \circ f)(x) =\)
Checkpoint 5.2.92 .
Given
\(f(x) = \sqrt{2x + 1}\) and
\(g(x) = x^2\text{,}\) \((f \circ g)(x) =\)
Checkpoint 5.2.93 .
Given
\(f(x) = \frac{1}{x - 3}\) and
\(g(x) = x + 2\text{,}\) \((f \circ g)(x) =\)
Checkpoint 5.2.94 .
Given
\(f(x) = x^3\text{,}\) \(g(x) = 2x\text{,}\) and
\(h(x) = x + 1\text{,}\) \((f \circ g \circ h)(x) =\)
Checkpoint 5.2.95 .
Given
\(f(x) = \sqrt{x}\) and
\(g(x) = 3x - 6\text{,}\) \((g \circ f)(x) =\)
Checkpoint 5.2.96 .
Given
\(f(x) = (x + 1)^2\) and
\(g(x) = \sqrt{x}\text{,}\) \((g \circ f)(x) =\)
Subsection 5.2.9 Asymptotes and Graphs of Functions
Checkpoint 5.2.97 .
Find the vertical asymptote of \( f(x) = \frac{x+2}{x-4} \).
Checkpoint 5.2.98 .
Find the horizontal asymptote of \( f(x) = \frac{3x^2 + 5}{x^2 - 1} \).
Checkpoint 5.2.99 .
Determine the slant (oblique) asymptote of \( f(x) = \frac{x^2 - x}{x + 1} \).
Checkpoint 5.2.100 .
At which value of \( x \) does \( f(x) = \ln(x-3) \) have a vertical asymptote?
Checkpoint 5.2.101 .
Identify the vertical asymptote(s) of \( f(x) = \frac{x-1}{x^2 - 1} \).
\( x = 1 \) and \( x = -1 \)
Checkpoint 5.2.102 .
Find the horizontal asymptote of \( f(x) = \frac{2x + 1}{x^2 + 4} \).
Checkpoint 5.2.103 .
Find the horizontal asymptote of \( f(x) = 5e^{-x} + 2 \).
Checkpoint 5.2.104 .
On the interval \([0, \pi]\), where is the vertical asymptote of \( f(x) = \sec(x) \)?
Checkpoint 5.2.105 .
Which feature describes the graph of \( f(x) = \frac{1}{x^2} \)?
Vertical asymptote at \( x=0 \) and horizontal at \( y=0 \)
Only a vertical asymptote at \( x=0 \)
Only a horizontal asymptote at \( y=0 \)
Checkpoint 5.2.106 .
Find the horizontal asymptote of \( f(x) = \frac{x^3 - 1}{2x^3 + 4x} \).
