91Ó°ÊÓ

Understanding the domain of a rational function is essential when graphing. A rational function has the form \( f(x) = \frac{P(x)}{Q(x)} \) where \( P(x) \) and \( Q(x) \) are polynomials. The domain consists of all the real numbers except for where the denominator, \( Q(x) \) equals zero, since division by zero is undefined.

Looking at our example, \( f(x) = \frac{1 - x^2}{x} \) has a denominator of \( x \). Therefore, the domain excludes \( x = 0 \) because it would make the denominator zero. Hence, the domain is all real numbers except zero, which we denote as \( x eq 0 \), or \( (-\infty, 0) \cup (0, \infty) \).

The intercepts of a function are points where the graph crosses the x-axis and y-axis. For the x-intercepts, we set the output of the function, \( f(x) \) to zero and solve for \( x \). This is because the points on the x-axis have a y-coordinate of zero. Conversely, to find the y-intercept, we set \( x \) to zero and solve for \( f(x) \) since the points on the y-axis have an x-coordinate of zero.

In our function \( f(x) = \frac{1 - x^2}{x} \), setting \( f(x) = 0 \) yields \( x \) values of 1 and -1, leading to two x-intercepts: \( (-1, 0) \) and \( (1, 0) \). As for the y-intercept, since setting \( x = 0 \) is not possible (the function is undefined at \( x = 0 \)), our graph has no y-intercept.

Slant or oblique asymptotes occur in rational functions when the degree of the numerator is one more than the degree of the denominator. They describe the end behavior of the function and appear as lines that the graph tends towards as \( x \) approaches \( -\infty \) or \( \infty \).

To find a slant asymptote, you can perform polynomial long division or synthetic division of the numerator by the denominator. In our function, \( f(x) = \frac{1 - x^2}{x} \), the numerator has a degree of 2 (since \( x^2 \) is the highest degree term) and the denominator has a degree of 1. Dividing \( 1 - x^2 \) by \( x \) yields a quotient of \( -x \) with a remainder. Thus, the slant asymptote is the line \( y = -x \) since the remainder becomes insignificant as \( x \) grows in magnitude.

Vertical asymptotes are lines that the graph of a function approaches but never touches. They indicate where the function heads towards infinity or negative infinity. For rational functions, vertical asymptotes occur at values of \( x \) that cause the denominator to be zero (as long as the numerator isn't also zero at these points, as that would suggest a hole instead).

For our function, the denominator is \( 0 \) when \( x = 0 \). Hence, there is a vertical asymptote at \( x = 0 \) because the function is undefined at this value. When graphing, you would draw a dashed line along \( x = 0 \) to represent this asymptote and show that the function increases or decreases without bound as it approaches this line.