In Exercises 81–88, a. Find the slant asymptote of the graph of each rational function and b. Follow the seven-step strategy and use the slant asymptote to graph each rational function. f(x)=(x²−1)/x

Verified step by step guidance

1. Identify the given rational function: $f\left(x\right)=\frac{x^2-1}{x}$.
2. To find the slant asymptote, perform polynomial long division of the numerator $x^2-\; 1$ by the denominator $x$.
3. Divide the leading term of the numerator $x^2$ by the leading term of the denominator $x$ to get the first term of the quotient: $x$.
4. Multiply the divisor $x$ by this quotient term $x$ and subtract the result from the numerator to find the remainder.
5. Express the function as $f\left(x\right)=x+\frac{remainder}{x}$. The slant asymptote is the linear function given by the quotient, which is $y\; =\; x$.

Key Concepts

Rational Functions

A rational function is a ratio of two polynomials, expressed as f(x) = P(x)/Q(x), where Q(x) ≠ 0. Understanding the behavior of rational functions, including their domains and discontinuities, is essential for analyzing their graphs and asymptotes.

Slant (Oblique) Asymptotes

Slant asymptotes occur when the degree of the numerator is exactly one more than the degree of the denominator. They represent the line that the graph approaches as x approaches infinity or negative infinity, found by performing polynomial division.

Graphing Rational Functions Using Asymptotes

Graphing rational functions involves identifying asymptotes, intercepts, and behavior near discontinuities. The seven-step strategy typically includes finding domain, intercepts, asymptotes (vertical, horizontal, or slant), and plotting points to sketch an accurate graph.