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)=x/(x^2+4)

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1. Identify the rational function given: $r\left(x\right)=\frac{x}{x^2}$. Note that the denominator is $x^2+4$.
2. To find vertical asymptotes and holes, first determine where the denominator is zero because these are the points where the function is undefined. Set the denominator equal to zero: $x^2+4=\; 0$.
3. Solve the equation $x^2\; +\; 4\; =\; 0$ for $x$. This involves isolating $x^2$ and then taking the square root of both sides.
4. Check if the solutions to the denominator equal zero are real numbers. If there are no real solutions, then there are no vertical asymptotes or holes because the function is defined for all real $x$.
5. Since the denominator does not factor into terms that cancel with the numerator, there are no holes. Summarize the findings about vertical asymptotes and holes based on the solutions obtained.

Key Concepts

Rational Functions

A rational function is a ratio of two polynomials, expressed as f(x) = P(x)/Q(x). Understanding the form of rational functions helps identify where the function is undefined, which is crucial for finding vertical asymptotes and holes.

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. They represent lines the graph approaches but never touches.

Holes in the Graph

Holes occur when a factor cancels out from both numerator and denominator, making the function undefined at that point but not causing an asymptote. The graph has a removable discontinuity at the x-value corresponding to the hole.