Problem 9 - Precalculus : Building Concepts and Connections

Question

Find the domain and the vertical and horizontal asymptotes (if any). $$g(x)=\frac{3}{x^{2}-4}$$

Step-by-Step Solution

Answer

The domain of \(g(x)=\frac{3}{x^{2}-4}\) is \(x \in (-\infty,-2) \cup (-2,2) \cup (2,\infty)\). The vertical asymptotes are at \(x = -2\) and \(x = 2\), and the horizontal asymptote is at \(y = 0\).

Step 1: Determine the domain

To find the domain of the function, \(g(x) = \frac{3}{x^{2}-4}\) must be determined. The denominator equals zero when \(x^{2}-4=0\), or \(x=\pm2\). So the domain is \(x \in (-\infty,-2) \cup (-2,2) \cup (2,\infty)\). In this case, 'domain' refers to all values x can take except those for which the function is undefined.

Step 2: Find the vertical asymptotes

Vertical asymptotes occur when the function becomes infinite, which generally occurs when the denominator equals zero. Here the denominator \(x^2 - 4\) equals zero when \(x = \pm 2\). Therefore, there are vertical asymptotes at \(x = -2\) and \(x = 2\).

Step 3: Find the horizontal asymptotes

If the degree of the polynomial in the denominator of a rational function is greater than the degree in the numerator (which it is here), then the x-axis (y=0) is a horizontal asymptote. So there is a horizontal asymptote at \(y = 0\).

Key Concepts

Domain of a Function

When we talk about the "domain of a function," we're really discussing the set of all possible input values (x-values) that the function can accept. For rational functions, these are usually all real numbers except where the denominator is zero.
In the given function, \(g(x) = \frac{3}{x^{2} - 4}\), the denominator, \(x^{2} - 4\), becomes zero when \(x = \pm 2\). Since division by zero is undefined, these values are excluded from the domain. Thus, the domain of \(g(x)\) is all real numbers except \(x = -2\) and \(x = 2\).
This is expressed as \(x \in (-\infty, -2) \cup (-2, 2) \cup (2, \infty)\). Understanding the domain is crucial; it tells us where our function can "live" on the number line.

Vertical Asymptotes

Vertical asymptotes in a function occur where the function's value heads towards infinity, usually corresponding to values that make the denominator zero. For our function, \(g(x) = \frac{3}{x^{2} - 4}\), we've already determined that \(x = -2\) and \(x = 2\) make the denominator zero.
As \(x\) approaches these values, the function \(g(x)\) grows infinitely large or small, creating lines at \(x = -2\) and \(x = 2\) which the graph comes infinitely close to but never actually crosses.

• Vertical asymptotes are depicted as dashed lines on a graph, symbolizing points of near-infinite behavior.
• In practical terms, this signifies that the function values shoot up or down very sharply at these x-values.

Vertical asymptotes are a hallmark of the areas where the function cannot be defined due to division by zero.

Horizontal Asymptotes

Horizontal asymptotes reveal the behavior of a function as \(x\) tends to infinity or negative infinity. They indicate a value that the function approaches but never quite reaches. For rational functions like \(g(x) = \frac{3}{x^{2} - 4}\), the form \(\frac{a}{b(x)}\) lets us identify horizontal asymptotes based on the relative degrees of the polynomial in the numerator and denominator.
Here, the degree of the denominator \(x^{2}\) (2) is greater than that of the numerator (0), leading us to conclude that the x-axis itself, \(y = 0\), acts as a horizontal asymptote.
Horizontal asymptotes suggest that:

• The function levels off, or stabilizes, as x-values become very large or very small.
• This behavior implies a fundamental limit to the function's values, regardless of how extreme the x-values grow.

Recognizing horizontal asymptotes helps in visualizing the long-term behavior of the graph of a function.