In Exercises 21–36, find the vertical asymptotes, if any, and the values of x corresponding to holes, if any, of the graph of each rational function. f(x)=x/(x+4)

Key Concepts

Rational Functions

A rational function is a function that can be expressed as the ratio of two polynomials. In the case of f(x) = x/(x+4), the numerator is x and the denominator is (x+4). Understanding the structure of rational functions is essential for analyzing their behavior, particularly in identifying asymptotes and holes.

Vertical Asymptotes

Vertical asymptotes occur in a rational function when the denominator approaches zero while the numerator does not simultaneously approach zero. For f(x) = x/(x+4), the vertical asymptote can be found by setting the denominator (x+4) equal to zero, leading to x = -4. This indicates that the function will approach infinity as x approaches -4.

Holes in the Graph

Holes in the graph of a rational function occur at values of x where both the numerator and denominator equal zero, indicating a removable discontinuity. In f(x) = x/(x+4), there are no holes since the numerator (x) does not equal zero when the denominator (x+4) does. Thus, identifying holes requires checking for common factors in the numerator and denominator.