# 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 included in this site.

## Questions with Solutions

## Question 1

**True or False**. The domain of a function is the set of all real values for which the function is real valued.

__Answer :__

True.

## Question 2

**True 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 3

**True 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 4

**True 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 \( + \infty \) or \( - \infty \) (very large or very small values).

## Question 5

**True 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 \( f(x) = \dfrac{x + 3}{x^
2 - 9} \).

Factor the denominator and simplify to obtain \( f(x) = \dfrac{1}{x - 3} \).

Although \( x = -3 \) makes the denominator equal to 0 there is no vertical asymptote at \( x = -3 \); in fact, there is a hole.

## Question 6

**True or False**. A horizontal asymptote may intersect the graph of the function.

__Answer :__

True.

Example: \( f(x) = \dfrac{\sin x}{x} \)

## Question 7

**True or False**. The x intercepts of the graph of a function correspond to the zeros of the function.

__Answer :__

True.
__Question 8:__

**True or False**. A graph cannot cut its vertical asymptote.

__Answer :__

True.

## References and Links

Calculus questions with answers and Calculus tutorials and problems .