Follow the seven steps to graph each rational function. f(x)=−x/(x+1)

Verified step by step guidance

1. Identify the function given: \(f(x) = \frac{-x}{x+1}\). This is a rational function where the numerator is \(-x\) and the denominator is \(x+1\).
2. Find the domain by determining where the denominator is zero. Set \(x+1=0\) and solve for \(x\) to find any vertical asymptotes or restrictions.
3. Find the vertical asymptote(s) by noting the values of \(x\) that make the denominator zero (from the domain step). These are the lines where the function is undefined.
4. Find the horizontal asymptote by comparing the degrees of the numerator and denominator. Since both numerator and denominator are degree 1, divide the leading coefficients to find the horizontal asymptote.
5. Find the intercepts: For the \(y\)-intercept, evaluate \(f(0)\). For the \(x\)-intercept, set the numerator equal to zero and solve for \(x\).

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 domain restrictions, where the denominator equals zero, is essential to avoid undefined values and to identify vertical asymptotes.

Asymptotes of Rational Functions

Asymptotes are lines that the graph approaches but never touches. Vertical asymptotes occur where the denominator is zero, and horizontal or oblique asymptotes describe end behavior as x approaches infinity or negative infinity.

Graphing Steps for Rational Functions

Graphing involves identifying domain restrictions, intercepts, asymptotes, and plotting points to understand the function's behavior. Following a systematic seven-step process ensures a complete and accurate graph of the rational function.