Find the vertical asymptotes, if any, and the values of x corresponding to holes, if any, of the graph of each rational function. g(x)=(x−3)/(x²−9)

Verified step by step guidance

1. Start by identifying the rational function given: \(g(x) = \frac{x - 3}{x^{2} - 9}\).
2. Factor the denominator to find values that make it zero: \(x^{2} - 9\) factors as \((x - 3)(x + 3)\).
3. Set the denominator equal to zero to find potential vertical asymptotes or holes: solve \((x - 3)(x + 3) = 0\), which gives \(x = 3\) and \(x = -3\).
4. Check if any factor in the numerator cancels with a factor in the denominator. Since the numerator is \(x - 3\), it cancels with the \((x - 3)\) factor in the denominator, indicating a hole at \(x = 3\).
5. The remaining factor in the denominator, \((x + 3)\), does not cancel, so \(x = -3\) is 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.