Problem 106 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the domain of the function and identify any horizontal or vertical asymptotes. $$f(x)=3-\frac{2}{x^{2}}$$

Step-by-Step Solution

Answer

The domain of the function \(f(x) = 3 - \frac{2}{x^{2}}\) is all real numbers except zero (\(- \infty, 0) ∪ (0, \infty)\). The function has a vertical asymptote at \(x = 0\) and a horizontal asymptote at \(y = 0\).

Step 1: Identifying the Domain

The given function is \(f(x) = 3 - \frac{2}{x^{2}}\). This function is defined for all real numbers except when \(x = 0\), as division by zero is undefined. Therefore, the domain of the function is all real numbers excluding zero. In interval notation, it is written as \((- \infty, 0) ∪ (0, \infty)\).

Step 2: Identifying Vertical Asymptotes

A vertical asymptote exists where a function becomes undefined. In the given function, \(f(x)\) becomes undefined when \(x = 0\). Therefore, we have a vertical asymptote at \(x = 0\).

Step 3: Identifying Horizontal Asymptotes

To find the horizontal asymptote, first look at the degrees of the numerator and denominator. Here, the degree of the numerator is 0 and the degree of the denominator is 2. Since the degree of the numerator is less than the degree of the denominator, the x-axis (y=0) is a horizontal asymptote.

Key Concepts

Vertical Asymptotes

Understanding vertical asymptotes is crucial when analyzing the behavior of rational functions, such as the function given: \(f(x) = 3 - \frac{2}{x^2}\). A vertical asymptote occurs where a function's value approaches infinity, thus becoming undefined.

• In rational functions, vertical asymptotes typically happen where the denominator is zero.
• In our function, the denominator is \(x^2\), which equals zero when \(x = 0\).
• Thus, a vertical asymptote exists at \(x = 0\).

This means as \(x\) approaches zero, \(f(x)\) will increase or decrease without bound. The graph of the function approaches the line \(x = 0\), creating this vertical barrier.

Horizontal Asymptotes

Horizontal asymptotes provide insight into the behavior of a function as \(x\) approaches positive or negative infinity. The horizontal asymptote helps us understand the end behavior of a function:

• For the function \(f(x) = 3 - \frac{2}{x^2}\), the degree of the numerator (which is zero, since there is no \(x\)) is less than the degree of the denominator (which is 2, because of \(x^2\)).
• In such cases, the horizontal asymptote is at \(y = 0\), which is the x-axis.

This indicates that as \(x\) grows larger or more negative, \(f(x)\) approaches zero but never fully reaches or crosses the x-axis. This is valuable for predicting the nature of the graph at extreme values of \(x\).

Interval Notation

Interval notation is a handy tool to express the domain of functions clearly and efficiently. For the function \(f(x) = 3 - \frac{2}{x^2}\), identifying when the function is not defined is crucial. Let’s see how interval notation helps:

• The domain of the function includes all real numbers except \(x = 0\), where the function is undefined.
• Using interval notation, we write this as \((-\infty, 0) \cup (0, \infty)\).
Interval notation thus provides a concise way to show that the domain spans all real numbers except a single point, making it a valuable tool for describing where the function is valid. It’s particularly useful when conveying complex domains in a standardized mathematical language.