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)=(x²+4x−21)/(x+7)

Verified step by step guidance

1. Start by identifying the rational function given: \(r(x) = \frac{x^{2} + 4x - 21}{x + 7}\).
2. Factor the numerator \(x^{2} + 4x - 21\) to see if any factors cancel with the denominator. To factor, find two numbers that multiply to \(-21\) and add to \$4$.
3. Write the factored form of the numerator and check if the denominator \(x + 7\) is a factor of the numerator. If it is, this indicates a hole at the value of \(x\) that makes \(x + 7 = 0\).
4. If the factor cancels, the hole occurs at \(x = -7\). If it does not cancel, then \(x = -7\) is a vertical asymptote because the denominator is zero there and the function is undefined.
5. Summarize: vertical asymptotes occur where the denominator is zero and not canceled by the numerator, and holes occur where a factor cancels between numerator and denominator.

Key Concepts

Rational Functions

A rational function is a ratio of two polynomials, expressed as f(x) = P(x)/Q(x). Understanding the behavior of rational functions involves analyzing their numerators and denominators, especially where the denominator equals zero, which affects the domain and graph.

Vertical Asymptotes

Vertical asymptotes occur at values of x where the denominator of a rational function is zero and the numerator is nonzero, causing the function to approach infinity or negative infinity. Identifying these points helps describe the function's behavior near undefined values.

Holes in the Graph

Holes occur when a factor cancels out from both numerator and denominator, indicating a removable discontinuity. At these x-values, the function is undefined, but the limit exists, resulting in a 'hole' rather than an asymptote on the graph.