The questions below are designed to help you develop a deep understanding of the properties of function graphs, which are crucial in Calculus. You may need to review definitions and theorems related to graphing functions. More on graphing techniques is included on this site.
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.
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