Problem 28 - Algebra and Trigonometry

Question

Find the vertical asymptotes, if any, and the values of \(x\) corresponding to holes, if any, of the graph of each crational function. $$ r(x)=\frac{x}{x^{2}+3} $$

Step-by-Step Solution

Answer

There are no vertical asymptotes and holes for the function \(r(x)=\frac{x}{x^{2}+3}\) in the domain of real numbers.

Step 1: Determine the Vertical Asymptotes

Set the denominator of the function equal to zero and solve for \(x\). In this case, the equation will be \(x^2 + 3 = 0\). Solving this, will yield that \(x\) cannot be real numbers thus there are no vertical asymptotes for real numbers.

Step 2: Find the Holes in the Function

Holes in the graph of a rational function occur when a factor in the numerator and a factor in the denominator cancel each other out. In this case of \(r(x) = \frac{x}{x^2 + 3}\), there are no common factors between the numerator and the denominator, so there are no holes.

Key Concepts

Vertical Asymptotes

Vertical asymptotes are a key feature of rational functions. They occur where the denominator of a rational function equals zero, and thus the function itself becomes undefined. Imagine dividing something by zero; mathematically, it simply doesn't compute.
To find the vertical asymptotes of a rational function like \( r(x)=\frac{x}{x^{2}+3} \), you start by setting the denominator equal to zero: \( x^2 + 3 = 0 \). Solving this will reveal if any real values for \( x \) cause the denominator to be zero.
In this case, the equation \( x^2 + 3 = 0 \) doesn't yield any real numbers, because \( x^2 = -3 \) is not possible with real numbers in our mathematical universe. Thus, there are no vertical asymptotes for this function on the real number line. Remember, vertical asymptotes are usually also vertical lines on a graph which the function approaches but never touches.

Holes in Graphs

Holes in the graph of a rational function offer another interesting insight. They occur when a factor in both the numerator and the denominator cancels each other out.
For the function \( r(x) = \frac{x}{x^2 + 3} \), finding holes involves looking for common factors in the numerator and denominator. If we had a situation like \( \frac{(x-1)(x+2)}{(x-1)(x-3)} \), the \( (x-1) \) factor in the numerator and denominator would cancel, creating a hole at \( x = 1 \).
However, in our given function, there are no common factors between \( x \) and \( x^2 + 3 \), meaning holes simply do not exist here. This is a key distinction that separates holes from vertical asymptotes, emphasizing how crucial factorization is in finding them.

Graph of Rational Functions

Understanding how to graph rational functions might seem like plotting a new terrain as each function has its unique characteristics.
When graphing a function like \( r(x) = \frac{x}{x^{2}+3} \), you'd investigate both vertical asymptotes and holes first, as they dictate the overall structure of the graph. In this case, there are neither. However, what you'd observe is how the function behaves as \( x \) approaches infinity or negative infinity. The absence of vertical asymptotes and holes makes the function smoother without interruptions.
To thoroughly comprehend the graph, it also helps to evaluate the function at selected values of \( x \), ensuring the shape of the graph is apparent. Features such as intercepts and end behavior form the rest of the picture, capturing the essence of what the graph of a rational function like \( r(x) \) represents.