Properties of the Graphs of Functions
The questions below have been designed to help you gain deep understanding of the properties of the graphs of functions which are of major importance in Calculus. You may need to review some definitions and theorems related to the graphs of functions in order to answer the questions below. More on graphing is located in this site.
Questions with SolutionsQuestion 1True or False. The domain of a function is the set of all real values for which the function is real valued.Answer : True.
Question 2True or False. The sign of the first derivative of a given function f informs you on the interval(s) where f(x) is positive, negative or equal to zero.Answer : False. The sign of the first derivative informs you on the interval(s) where f is increasing, decreasing or constant.
Question 3True or False. The sign of the second derivative of a given function f informs you on the concavity of the graph of f.Answer : True.
Question 4True or False. The horizontal asymptote to the graph of a given function f is determined by finding the limit, if it exists, of f(x) as x approaches 0.Answer : False. A horizontal asymptote may be determined by finding the limit of f(x) as x approaches + or  infinity (very large or very small values).
Question 5True or False. Any value of x that makes the denominator of rational function f equal to zero, represents a vertical asymptote to the graph of f.Answer : False. Not always. Let f(x) = (x + 3) / (x^{ 2} 9). Factor the denominator and simplify to obtain f(x) = 1 / (x  3) Although x =  3 makes the denominator equal to 0 there is no vertical asymptote at x =  3; in fact there is a hole.
Question 6True or False. A horizontal asymptote may intersect the graph of the function.Answer : True. Example: f(x) = sin x / x
Question 7True or False. The x intercepts of the graph of a function corresponds to the zeros of the function.Answer : True.
Question 8:
More references on calculus questions with answers and tutorials and problems .
