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)=(x²+4x−21)/(x+7)

Verified step by step guidance

1. Identify the rational function given: $r\left(x\right)=\frac{x^2+4x-21}{x}$.
2. Factor the numerator to see if any factors cancel with the denominator. Factor $x^2+\; 4x\; -\; 21$ by finding two numbers that multiply to -21 and add to 4.
3. Rewrite the rational function with the factored numerator and the denominator: $r\left(x\right)=\frac{(x\; +\; ?)(x\; +\; ?)}{x}$. Identify if any factor cancels with the denominator.
4. Determine vertical asymptotes by setting the denominator equal to zero and solving for $x$, excluding any values where factors cancel (which would indicate holes instead).
5. Identify holes by finding values of $x$ that make both numerator and denominator zero (common factors), and state these as points where the graph has holes.

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.