The questions below are designed to help you develop a deep understanding of the properties of function graphs, which are crucial in Calculus[cite: 1]. You may need to review definitions and theorems related to graphing functions. More on graphing techniques is included on this site. Each solution is hidden inside a collapsible dropdown so you can practice independently before reviewing the steps.
True or False: The domain of a function is the set of all real values for which the function is real-valued.
Answer: True.
True or False: The sign of the first derivative of a function \( f \) informs you on the interval(s) where \( f(x) \) is positive, negative, or zero.
Answer: False.
The sign of the first derivative informs you about the interval(s) where \( f \) is increasing, decreasing, or constant.
True or False: The sign of the second derivative of a function \( f \) informs you about the concavity of the graph of \( f \).
Answer: True.
True or False: The horizontal asymptote of a function \( f \) is determined by finding \(\lim_{x \to 0} f(x)\).
Answer: False.
A horizontal asymptote is determined by \(\lim_{x \to +\infty} f(x)\) or \(\lim_{x \to -\infty} f(x)\).
True or False: Any value of \( x \) that makes the denominator of a rational function \( f \) zero represents a vertical asymptote.
Answer: False.
For example, consider:
\[
f(x) = \frac{x + 3}{x^2 - 9} = \frac{1}{x - 3} \quad \text{after simplification}.
\]
Although \( x = -3 \) makes the denominator zero, there is no vertical asymptote there; it is a hole in the graph.
True or False: A horizontal asymptote may intersect the graph of the function.
Answer: True.
Example:
\[
f(x) = \frac{\sin x}{x}.
\]
True or False: The x-intercepts of a function correspond to its zeros.
Answer: True.
True or False: A graph cannot cut its vertical asymptote.
Answer: True.
Calculus questions with answers[cite: 1] and Calculus tutorials and problems[cite: 1].