Problem 22 - Algebra and Trigonometry

Question

Find the vertical asymptotes, if any, and the values of \(x\) corresponding to holes, if any, of the graph of each crational function. $$ f(x)=\frac{x}{x-3} $$

Step-by-Step Solution

Answer

The vertical asymptote of the function \(f(x) = \frac{x}{x - 3}\) is at \(x = 3\). The function does not have any holes.

Step 1: Identifying Potential Vertical Asymptotes

The vertical asymptotes of a rational function are the values of \(x\) for which the denominator of the function equals zero, yet are not common factors with the numerator. Therefore, to find the vertical asymptotes, solve the equation \(x-3=0\) which gives \(x=3\).

Step 2: Identifying Holes

Holes in the graph of a rational function occur when there are common factors between the numerator and the denominator. However, no common factors exist within \(f(x)=\frac{x}{x-3}\) besides 1. Therefore, there are no holes in the graph.

Key Concepts

Rational Function

A rational function is defined as a function that can be expressed as the quotient of two polynomials, i.e., in the form of \( f(x) = \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials and \( Q(x) \) is not equal to zero. The domain of a rational function is all real numbers except for where the denominator \( Q(x) \) is equal to zero.

These functions are useful to model scenarios in various fields such as engineering, physics, and economics, where the relationship between two quantities can be expressed as a ratio. Being able to find vertical asymptotes and holes, as well as graph the function is crucial for understanding the behavior of these ratios in different contexts.

Asymptote of Rational Function

An asymptote of a rational function refers to a line that the graph of the function approaches, but never actually reaches. The most common type of asymptote encountered with rational functions is the vertical asymptote, which occurs perpendicular to the x-axis. To find vertical asymptotes, one must look at the values that would make the denominator equal to zero. After determining these values, we must check if they are cancelled out by factors in the numerator.

For the function \( f(x) = \frac{x}{x-3} \), solving \( x-3 = 0 \) suggests that \( x=3 \) could be where a vertical asymptote lies. Since this value does not nullify the numerator, the vertical asymptote is indeed at \( x=3 \).

Holes in Rational Functions

Holes in a graph of a rational function are points where the function is not defined due to a common factor in the numerator and denominator that cancels out. They are represented as a single point on the graph where the function does not exist. Unlike asymptotes which describe a trend of the graph over an infinite section, holes are punctual absences in an otherwise continuous curve.

However, for the function \( f(x) = \frac{x}{x-3} \), we see that there are no common factors between the numerator and denominator that cancel each other; thus, no holes are present. In other functions, to find a hole, one would factorize both the numerator and denominator and simplify common terms, then set the denominator equal to zero and solve for \( x \).

Graphing Rational Functions

Graphing rational functions is a key skill for visualizing their behavior. It involves several steps beyond plotting points, such as identifying asymptotes and holes, as mentioned earlier. The first step is to determine the overall shape by finding the x- and y-intercepts, asymptotes, and holes. Next, the function should be plotted over a range of values, paying special attention near the asymptotes and holes.

To accurately graph \( f(x) = \frac{x}{x-3} \), we would plot a vertical asymptote at \( x=3 \) and analyze the behavior of \( f(x) \) as \( x \) gets closer to 3 from both the left and the right. We should also note the horizontal intercept at (0,0) and the fact that there are no holes in this particular function. By doing so, we can get a clear picture of the function's characteristics.