Problem 10 - A Graphical Approach to Precalculus with Limits

Question

Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of each rational function. State the domain of \(f .\) $$f(x)=\frac{-6}{x+9}$$

Step-by-Step Solution

Answer

Vertical asymptote at \( x=-9 \), horizontal asymptote at \( y=0 \); domain: all real numbers except \( x = -9 \).

Step 1: Identify Asymptotes

The given function is \( f(x) = \frac{-6}{x+9} \). The function is rational, written as \( \frac{N(x)}{D(x)} \) with \( N(x) = -6 \) and \( D(x) = x+9 \).For vertical asymptotes, set \( D(x) = 0 \). Therefore, solve \( x+9=0 \).Hence, the vertical asymptote is at \( x = -9 \).For horizontal asymptotes, compare the degrees of the numerator and the denominator. The degree of the numerator is 0 and the degree of the denominator is 1.Therefore, the horizontal asymptote is at \( y = 0 \).There are no oblique asymptotes as the numerator's degree is not one less than the denominator's.

Step 2: Determine Domain

The function \( f(x) = \frac{-6}{x+9} \) is undefined where the denominator is zero, i.e., \( x = -9 \).Thus, the domain of \( f(x) \) includes all real numbers except \( x = -9 \). This can be written as \( \{x \in \mathbb{R} \mid x eq -9\} \).

Key Concepts

Vertical Asymptotes

When dealing with rational functions, vertical asymptotes occur where the denominator equals zero, and the function cannot be evaluated. Essentially, these are the places where the function shoots off to infinity. In our specific example, the function given is \( f(x) = \frac{-6}{x+9} \). To find the vertical asymptote, we set the denominator \( x+9 \) equal to zero and solve for \( x \). This gives us:

• \( x + 9 = 0 \)
• Thus, \( x = -9 \)

Therefore, there is a vertical asymptote at \( x = -9 \). This means that as \( x \) approaches \( -9 \), the value of \( f(x) \) either increases or decreases without bound. Recognizing vertical asymptotes is crucial for understanding where the function is not defined.

Horizontal Asymptotes

Horizontal asymptotes in rational functions indicate the behavior of the function as the input \( x \) becomes very large, either positively or negatively. These asymptotes tell us what the value of \( f(x) \) will approach as \( x \) reaches infinity. The function \( f(x) = \frac{-6}{x+9} \) has a numerator with a degree of 0 and a denominator with a degree of 1.

• If the degree of the numerator is less than the degree of the denominator, like here, the horizontal asymptote is always \( y = 0 \).
• If the degrees are the same, the horizontal asymptote will be the ratio of the leading coefficients.
• If the numerator's degree is greater, no horizontal asymptote exists. Instead, there might be an oblique asymptote.

In this case, the horizontal asymptote is at \( y = 0 \), meaning the function approaches \( y = 0 \) when \( x \) is very large or very small.

Domain of a Function

To fully comprehend a rational function, understanding its domain is crucial. The domain of a function is essentially all the possible values of \( x \) that you can plug into the function without causing any mathematical errors, such as division by zero. For the function \( f(x) = \frac{-6}{x+9} \), it is important to exclude any \( x \) values that make the denominator zero.

• Find the values where the denominator is zero: \( x+9 = 0 \).
• From this, we know that \( x = -9 \) is not in the domain because it would make the function undefined.

Thus, the domain of \( f(x) \) includes all real numbers except \( x = -9 \).This can be formally written as:

• \( \{x \in \mathbb{R} \mid x eq -9\} \)

Understanding the domain helps avoid undefined points in the function, ensuring you work within the valid range of the function's definition.