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. r(x)=(x^2+4x−21)/(x+7)

Key Concepts

Rational Functions

A rational function is a function that can be expressed as the ratio of two polynomials. In the given function r(x) = (x^2 + 4x - 21) / (x + 7), the numerator and denominator are both polynomials. Understanding the structure of rational functions is essential for analyzing their behavior, including 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 the function r(x), we find vertical asymptotes by setting the denominator (x + 7) equal to zero, leading to x = -7. This indicates that the function will approach infinity as x approaches -7.

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. To find holes, we factor both the numerator and denominator and identify common factors. In r(x), if the numerator can be factored to include (x + 7), it would indicate a hole at x = -7, which must be checked against the simplified function.