Problem 21 - 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. $$f(x)=\frac{x}{x+4}$$

Step-by-Step Solution

Answer

The function \(f(x) = \frac{x}{x+4}\) has a vertical asymptote at \(x = -4\) and it has no holes.

Step 1: Identify the form of the function

Here the given function is \(f(x)=\frac{x}{x+4}\). This is in the form of a rational function where numerator is \(x\) and denominator is \(x+4\). Both are polynomials, hence we can proceed with finding vertical asymptotes and holes.

Step 2: Find the vertical asymptotes

For a rational function, the vertical asymptotes occur where the denominator is zero. Set the denominator equal to zero and solve for \(x\). So, solve the equation \(x+4=0\). We find \(x=-4\). Therefore, \(x=-4\) is the vertical asymptote.

Step 3: Determine the holes

A hole occurs when both the numerator and denominator become zero at a particular \(x\) value. Checking the given function, there's no value of \(x\) which will make both \(x\) and \(x + 4\) equal to zero simultaneously. Thus, there are no holes in the given function.