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. f(x)=x/(x+4)

Verified step by step guidance

1. Identify the rational function: $f(x)=\frac{x}{x+4}$.
2. Determine the vertical asymptotes by setting the denominator equal to zero: $x+4=0$.
3. Solve for $x$ to find the vertical asymptote: $x=-4$.
4. Check for holes by factoring the numerator and denominator to see if there are any common factors. In this case, there are no common factors, so there are no holes.
5. Conclude that the vertical asymptote is at $x=-4$ and there are no holes in the graph of the function.

Key Concepts

Rational Functions

A rational function is a function that can be expressed as the ratio of two polynomials. In the case of f(x) = x/(x+4), the numerator is x and the denominator is (x+4). Understanding the structure of rational functions is essential for analyzing their behavior, particularly in 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 f(x) = x/(x+4), the vertical asymptote can be found by setting the denominator (x+4) equal to zero, leading to x = -4. This indicates that the function will approach infinity as x approaches -4.

Holes in the Graph

Holes in the graph of a rational function occur at values of x where both the numerator and denominator equal zero, indicating a removable discontinuity. In f(x) = x/(x+4), there are no holes since the numerator (x) does not equal zero when the denominator (x+4) does. Thus, identifying holes requires checking for common factors in the numerator and denominator.