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)/(x²+4x−21)

Verified step by step guidance

1. Start by identifying the denominator of the rational function, which is \(x^{2} + 4x - 21\).
2. Factor the denominator to find its roots. To factor \(x^{2} + 4x - 21\), look for two numbers that multiply to \(-21\) and add to \(4\).
3. Write the denominator as a product of two binomials based on the factors found: \(x^{2} + 4x - 21 = (x + a)(x + b)\), where \(a\) and \(b\) are the numbers from the previous step.
4. Set each factor equal to zero to find the values of \(x\) that make the denominator zero: \(x + a = 0\) and \(x + b = 0\). These values are potential vertical asymptotes or holes.
5. Check if any of these \(x\)-values also make the numerator zero. If a value makes both numerator and denominator zero, it corresponds to a hole; if it only makes the denominator zero, it corresponds to a vertical asymptote.

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 but the numerator is not zero, 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, resulting in a removable discontinuity. At these x-values, the function is undefined, but the limit exists, indicating a 'hole' rather than an asymptote on the graph.