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: \(f(x) = \frac{x^{2} + 1}{x}\).
2. To find the slant (oblique) asymptote, perform polynomial long division of the numerator by the denominator: divide \(x^{2} + 1\) by \(x\).
3. Set up the division: \(x\) divides into \(x^{2}\) exactly \(x\) times. Multiply \(x\) by \(x\) to get \(x^{2}\), subtract this from \(x^{2} + 1\) to find the remainder.
4. The remainder after subtracting is \$1\(. So, the division gives \)x\( with a remainder of \)1\(, which can be written as \)f(x) = x + \frac{1}{x}$.
5. The slant asymptote is the quotient without the remainder term, so it is the line \(y = x\). This line describes the behavior of \(f(x)\) as \(x\) approaches infinity or negative infinity.

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 asymptotes, is essential for graphing and analyzing their properties.

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.

Polynomial Division

Polynomial division is a method used to divide one polynomial by another, similar to long division with numbers. It helps find the quotient and remainder, which are used to determine slant asymptotes and simplify rational functions for graphing.