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. h(x)=(x+7)/(x^2+4x−21)

Key Concepts

Rational Functions

A rational function is a function that can be expressed as the ratio of two polynomials. In the given function h(x) = (x + 7) / (x^2 + 4x - 21), the numerator is a linear polynomial, and the denominator is a quadratic polynomial. 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. To find vertical asymptotes, we set the denominator equal to zero and solve for x. In this case, identifying the values of x that make the denominator zero will reveal the locations of any vertical asymptotes in the graph of h(x).

Holes in the Graph

Holes in the graph of a rational function occur at values of x that make both the numerator and denominator equal to zero, indicating a removable discontinuity. To find holes, we need to factor both the numerator and denominator and identify common factors. If a common factor exists, the x-value corresponding to that factor represents a hole in the graph, where the function is undefined but can be 'filled in' by simplifying the function.