Find the vertical asymptotes, if any, and the values of x corresponding to holes, if any, of the graph of each rational function. h(x)=x/x(x+4)

Verified step by step guidance

1. Start by simplifying the given rational function: \(h(x) = \frac{x}{x(x+4)}\). Notice that the numerator and denominator both contain the factor \(x\).
2. Cancel the common factor \(x\) from the numerator and denominator, but remember that \(x \neq 0\) because division by zero is undefined. After cancellation, the simplified function is \(h(x) = \frac{1}{x+4}\), with the restriction \(x \neq 0\).
3. Identify the vertical asymptotes by setting the denominator of the simplified function equal to zero: solve \(x + 4 = 0\) to find \(x = -4\). This value corresponds to a vertical asymptote because the function is undefined there and the factor was not canceled.
4. Determine if there are any holes by looking at the factors canceled during simplification. Since \(x\) was canceled, and \(x = 0\) makes the original denominator zero, there is a hole at \(x = 0\).
5. Summarize: the function has a vertical asymptote at \(x = -4\) and a hole at \(x = 0\).

Key Concepts

Rational Functions

A rational function is a ratio of two polynomials, expressed as f(x) = P(x)/Q(x). Understanding the form and behavior of rational functions is essential for analyzing their graphs, including identifying discontinuities such as vertical asymptotes and holes.

Vertical Asymptotes

Vertical asymptotes occur at values of x where the denominator of a rational function is zero but the numerator is not zero, causing the function to approach infinity or negative infinity. These represent lines where the graph grows without bound and the function is undefined.

Holes in the Graph

Holes occur when a factor cancels out from both numerator and denominator, resulting in a removable discontinuity. At these x-values, the function is undefined, but the limit exists, and the graph has a 'gap' rather than an asymptote.