Question

Identify the horizontal and vertical asymptotes, if any, of the given function. $$ f(x)=\frac{x^{2}-9}{9 x+27} $$

Short answer

There are no vertical or horizontal asymptotes; there is a hole at \( x = -3 \).

Step-by-step solution

Identify Potential Vertical Asymptotes

Vertical asymptotes occur when the denominator of a function is zero, but the numerator is not zero at the same value. To find potential vertical asymptotes for the function \( f(x)=\frac{x^{2}-9}{9x+27} \), set the denominator equal to zero: \( 9x + 27 = 0 \). Solve for \( x \) to find the location of the asymptote.

Solve for Vertical Asymptotes

Solve the equation from the previous step: \( 9x + 27 = 0 \). Subtract 27 from both sides to obtain: \( 9x = -27 \). Divide both sides by 9 to solve for \( x \): \( x = -3 \). Thus, there is a vertical asymptote at \( x = -3 \).

Simplify the Function Expression

Before finding horizontal asymptotes, check if the function can be simplified. Factor the numerator \( x^{2} - 9 \) as \( (x-3)(x+3) \). The function becomes \( f(x) = \frac{(x-3)(x+3)}{9(x+3)} \). Cancel the common factor \( (x+3) \) to simplify the function to \( f(x) = \frac{x-3}{9} \).

Understand the Effect of Simplification on Asymptotes

The simplification indicates there is no vertical asymptote at \( x = -3 \) since the factor \( (x+3) \) cancels out. Instead, at \( x = -3 \), there is a removable discontinuity (hole). Given the simplification to \( f(x) = \frac{x-3}{9} \), there are no vertical asymptotes.

Find Horizontal Asymptotes

Horizontal asymptotes are found by analyzing the limits of the function as \( x \) approaches infinity. For rational functions, compare the degree of the numerator (2) with the degree of the denominator (1). Since the degree of the numerator is greater than the degree of the denominator, the function does not have horizontal asymptotes.

Key concepts

Rational Functions

Rational functions are an essential part of algebra, and they can often be seen as a fraction of two polynomials. The general form is \( \frac{P(x)}{Q(x)} \) where both \( P(x) \) and \( Q(x) \) are polynomials. Most of the interesting behavior in rational functions happens at certain values where these fractions become undefined or behave differently, known as asymptotes and discontinuities.
Understanding rational functions involves recognizing the behavior both as \( x \) gets very large and at certain specific values that cause the denominator to equal zero. In this example, our rational function is \( f(x) = \frac{x^{2}-9}{9x+27} \). Observing how this function behaves helps us gain insight into the nature of asymptotes and any discontinuities present.
Students should become comfortable identifying the parts of a rational function and using them to predict behavior.

Vertical Asymptote

A vertical asymptote is a line \( x = a \), where the function becomes undefined, and the function's value approaches infinity or negative infinity as \( x \) approaches \( a \). Finding these lines involves setting the denominator equal to zero, since division by zero is undefined.
In our function \( f(x)=\frac{x^{2}-9}{9x+27} \), setting the denominator \( 9x+27 \) equal to zero gives us \( 9x+27=0 \). Solving for \( x \) results in \( x=-3 \), suggesting a vertical asymptote at this point.
However, simplification can sometimes change this conclusion, as we'll see with discontinuities.

Horizontal Asymptote

Horizontal asymptotes are lines \( y = b \) that the graph of the function approaches as \( x \) goes to infinity or negative infinity. These often provide a boundary for the graph at extreme positive or negative values of \( x \).
Determining horizontal asymptotes in rational functions involves comparing the degree of the numerator to the degree of the denominator. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is the x-axis \( y=0 \). If they are equal, the asymptote is the ratio of the leading coefficients. Because the numerator of our example, \( x^2 - 9 \), is of higher degree than the denominator \( 9x + 27 \), there is no horizontal asymptote in this case.
It's crucial to spot this, as it helps predict the end behavior of the function's graph.

Simplification

Simplification makes rational functions easier to interpret by reducing the expression to its simplest form. This often involves factoring polynomials in the numerator and denominator and canceling common factors.
For \( f(x)=\frac{x^{2}-9}{9x+27} \), factoring the numerator gives \( (x-3)(x+3) \). The function becomes \( \frac{(x-3)(x+3)}{9(x+3)} \). Canceling the \( (x+3) \) gives \( f(x)=\frac{x-3}{9} \).
Simplifying in this way can change our understanding of vertical asymptotes and reveal hidden behaviors in the function, like removable discontinuities.

Discontinuity

Discontinuities are points where the function is not continuous, meaning there is a break or hole in the graph. These can often be resolved by simplification, revealing points where a graph is simply undefined, rather than shooting off to infinity.
In our example, the discontinuity at \( x = -3 \) is initially thought to be a vertical asymptote. But after simplification, we recognize this point as a removable discontinuity, or a "hole," in the function \( f(x)=\frac{x-3}{9} \).
Understanding discontinuities is important to fully grasp the function's complete graph and behavior, emphasizing the significance of simplification.