Problem 23 - 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. $$g(x)=\frac{x+3}{x(x+4)}$$

Step-by-Step Solution

Answer

The function \(g(x)=\frac{x+3}{x(x+4)}\) has vertical asymptotes at \(x = 0\) and \(x = -4\). There are no holes in the function's graph.

Step 1: Find the Denominator

Firstly, we need to find the denominator of the given rational function. The denominator for the function \(g(x)=\frac{x+3}{x(x+4)}\) is \(x(x+4)\)

Step 2: Finding values that make the denominator zero

Next, we set the denominator equal to zero and solve the equation: \[x * (x + 4) = 0\] This equation is true if \(x = 0\) or \(x = -4\). These values might result in vertical asymptotes or holes, depending on whether they are also roots of the numerator expression.

Step 3: Checking for roots in the numerator

We check if the values of \(x\) that result in a zero denominator are also roots of the numerator. In this case, \(x = -3\) makes the numerator 0, but since neither 0 nor -4 are equal to -3, these numbers do not result in holes.

Step 4: Defining the Asymptotes

Finally, as neither of the values \(x = 0\) or \(x = -4\) result in 0 in the numerator, they are not hole-presenting values, but they are vertical asymptotes. Thus the vertical asymptotes of the function \(g(x)\) are \(x = 0\) and \(x = -4\).