In Exercises 21–36, 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)=(x+3)/x(x+4)

Verified step by step guidance

1. Identify the rational function: \( g(x) = \frac{x+3}{x(x+4)} \).
2. Determine the values of \( x \) that make the denominator zero, as these are potential vertical asymptotes or holes. Set \( x(x+4) = 0 \) and solve for \( x \).
3. Solve the equation \( x(x+4) = 0 \) to find \( x = 0 \) and \( x = -4 \).
4. Check if these values also make the numerator zero. If a value makes both the numerator and denominator zero, it corresponds to a hole. Otherwise, it is a vertical asymptote.
5. Since \( x = 0 \) and \( x = -4 \) do not make the numerator zero, these values correspond to vertical asymptotes.

Key Concepts

Rational Functions

A rational function is a function that can be expressed as the ratio of two polynomials. In the given function g(x) = (x + 3) / (x(x + 4)), the numerator is a polynomial of degree 1, and the denominator is a polynomial of degree 2. Understanding the structure of rational functions is essential for analyzing their behavior, including identifying asymptotes and holes.

Vertical Asymptotes

Vertical asymptotes occur in a rational function when the denominator approaches zero while the numerator does not simultaneously approach zero. For g(x), we find vertical asymptotes by setting the denominator x(x + 4) equal to zero, leading to the values of x that cause the function to be undefined. These values indicate where the graph will approach infinity.

Holes in the Graph

Holes in the graph of a rational function occur at values of x that make both the numerator and denominator equal to zero, indicating a removable discontinuity. In g(x), if the numerator (x + 3) shares a common factor with the denominator, it will create a hole. Identifying these values is crucial for accurately sketching the graph and understanding its behavior.