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. g(x)=(x−3)/(x²−9)

Verified step by step guidance

1. Identify the rational function given: $g\left(x\right)=\frac{x-3}{x^2-9}$.
2. Factor the denominator to find values of $x$ that make the denominator zero: $x^2-9\; =\; (x-3)(x+3)$.
3. Check for common factors in numerator and denominator. Since numerator is $x-3$ and denominator has a factor $x-3$, this indicates a hole at $x=3$.
4. After canceling the common factor $x-3$, the simplified function is for $x\; eq\; 3$.
5. Determine vertical asymptotes by setting the denominator of the simplified function equal to zero: $x+3=0$ gives $x=-3$, which 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 domain restrictions caused by the denominator being zero is essential, as these points often indicate vertical asymptotes or holes in the graph.

Vertical Asymptotes

Vertical asymptotes occur at values of x where the denominator is zero but the numerator is not zero, causing the function to approach infinity or negative infinity. Identifying these points helps describe the behavior of the graph near undefined values.

Holes in the Graph

Holes occur when a factor cancels out from both numerator and denominator, resulting in a removable discontinuity. The x-value causing the zero denominator and numerator factor corresponds to the hole, where the function is undefined but the limit exists.