Problem 27 - 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. $$r(x)=\frac{x}{x^{2}+4}$$

Step-by-Step Solution

Answer

The rational function \(r(x) = \frac{x}{x^2+4}\) has neither holes nor vertical asymptotes.

Step 1: Identify the denominator and solve for zero

To identify where the function may be undefined, solve the equation \(x^2 + 4 = 0\). Unfortunately, there are no real solutions to this equation because any squared number will be positive and never be able to sum with 4 to equals zero.

Step 2: Check any cancellations in simplified function

Now, look for any factors that cancel out in the original form of the function and its simplest form. If a factor cancels out, this would correspond to a hole at that x value. Since \(\frac{x}{x^2 + 4}\) is already simplified and there is no factor that can be cancelled, there are no removable discontinuities or holes present.

Step 3: Conclusion

Based on our calculations, the function \(r(x) = \frac{x}{x^2+4}\) has no vertical asymptotes and no holes.