Problem 28 - College Algebra

Question

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

Step-by-Step Solution

Answer

The function \( r(x)=\frac{x}{x^{2}+3} \) has neither vertical asymptotes nor holes.

Step 1: Identify the denominator of the function

In the given rational function, \( r(x)=\frac{x}{x^{2}+3} \), the denominator polynomial is \( x^{2} + 3 \).

Step 2: Find the vertical asymptotes

Vertical asymptotes occur at values of \(x\) where the denominator is zero. Here, \(x^{2} + 3 = 0\) solves to \( x^{2} = -3 \). Since the square of a real number is always non-negative, the equation has no real solutions. Therefore, the function \( r(x)=\frac{x}{x^{2}+3} \) has no vertical asymptotes.

Step 3: Identify any potential holes

Holes occur at points in \(x\) where both the numerator and the denominator become zero. However, in this function, at no point of \(x\) does both the numerator ( which is \(x\) ) and the denominator (which is \(x^{2} + 3\)) become zero together. So there are no holes in this function.