Problem 21 - Algebra 2

Question

Graph each rational function. $$ f(x)=\frac{x}{x-3} $$

Step-by-Step Solution

Answer

The graph of \( f(x) = \frac{x}{x-3} \) has a vertical asymptote at \( x = 3 \) and a horizontal asymptote at \( y = 1 \).

Step 1: Identify the Function's Domain

To find the domain of the function \( f(x) = \frac{x}{x-3} \), we need to identify any values of \( x \) that make the denominator zero, as division by zero is undefined. Setting the denominator equal to zero, we have \( x - 3 = 0 \). Solving this, we find \( x = 3 \). Thus, the domain of the function is all real numbers except \( x = 3 \).

Step 2: Locate Vertical Asymptotes

Vertical asymptotes occur where the denominator of the rational function becomes zero and the numerator does not cancel it out. For \( f(x) = \frac{x}{x-3} \), the vertical asymptote is at \( x = 3 \), as this is where the denominator becomes zero.

Step 3: Determine Horizontal Asymptotes

To find the horizontal asymptote, compare the degrees of the numerator and the denominator. Since both the numerator \( x \) and the denominator \( x-3 \) are of degree 1, the horizontal asymptote is found by dividing the leading coefficients: \( \frac{1}{1} = 1 \). Thus, the horizontal asymptote is \( y = 1 \).

Step 4: Sketch the Function

Begin by plotting the vertical asymptote (a dashed line) at \( x = 3 \) and the horizontal asymptote \( y = 1 \). Choose sample points, such as \( x = 0 \), \( x = 1 \), and \( x = 4 \), to calculate \( f(x) \) and get corresponding points. \( f(0) = 0 \), \( f(1) = \frac{1}{-2} \), and \( f(4) = 4 \). Connect these points, ensuring the curve approaches the asymptotes as it extends towards infinity.

Key Concepts

Vertical Asymptotes

A vertical asymptote of a rational function occurs at a value of \( x \) where the denominator is zero but the numerator is not. This results in the function approaching infinity, creating a line that the graph gets infinitely closer to but never actually touches. In the given function \( f(x) = \frac{x}{x-3} \), the vertical asymptote is located at \( x = 3 \).
This is because if we set the denominator \( x-3 = 0 \), we solve to get \( x = 3 \). It is important to note that this line is not part of the function's graph, it is rather a boundary where the function behavior changes dramatically.

• Remember, a vertical asymptote will never be crossed by the graph of a rational function.
• This is because the function is undefined at this point, causing a gap in the graph.

Understanding vertical asymptotes helps in predicting the behavior of functions and their graphs, especially near points of disruption.

Horizontal Asymptotes

Horizontal asymptotes of a rational function give us an idea of the end behavior of the graph. These occur when the output value \( y \) approaches a specific number as \( x \) tends towards either positive or negative infinity. For \( f(x) = \frac{x}{x-3} \), we find the horizontal asymptote by comparing the degrees of the numerator and the denominator.
Both the numerator \( x \) and the denominator \( x-3 \) are degree 1 polynomials. Thus, we simply take the leading coefficients' ratio, yielding \( \frac{1}{1} = 1 \). This means the horizontal asymptote is \( y = 1 \).

• It is possible for a graph to cross a horizontal asymptote.
• The asymptote merely dictates the direction the graph heads as it stretches into infinity.

Horizontal asymptotes play a crucial role in understanding graph tendencies, allowing predictions about the graph's progression away from the center.

Function Domain

The domain of a function is the set of all possible \( x \) values that can be input into the function without causing mathematical inconsistencies. For a rational function such as \( f(x) = \frac{x}{x-3} \), determining the domain involves identifying values of \( x \) that make the denominator equal zero, as this creates an undefined expression.
By setting \( x-3 = 0 \), we find that \( x = 3 \) would cause such an issue.

• Every value except \( x = 3 \) can be used in the function, which defines its domain as all real numbers except that value.
• Knowing the domain is essential in graphing as it tells us where the function is not applicable.

Comprehending the domain of rational functions aids in avoiding undefined values and ensuring accurate graph representations.