Follow the seven steps to graph each rational function. f(x)=2x²/(x²−1)

Verified step by step guidance

1. Identify the domain of the function by finding the values of \(x\) that make the denominator zero. Solve the equation \(x^{2} - 1 = 0\) to find these values, since the function is undefined there.
2. Find the vertical asymptotes by setting the denominator equal to zero and solving for \(x\). These are the lines where the function approaches infinity or negative infinity.
3. Determine the horizontal or oblique asymptote by comparing the degrees of the numerator and denominator. Since both numerator and denominator are degree 2, find the horizontal asymptote by dividing the leading coefficients.
4. Find the intercepts: For the \(y\)-intercept, evaluate \(f(0)\). For the \(x\)-intercepts, set the numerator equal to zero and solve for \(x\).
5. Analyze the behavior of the function near the vertical asymptotes and at points in each interval determined by the vertical asymptotes to understand how the graph behaves on those intervals.

Key Concepts

Rational Functions

A rational function is a ratio of two polynomials, expressed as f(x) = P(x)/Q(x). Understanding its domain, where the denominator Q(x) ≠ 0, is essential to avoid undefined points and to analyze behavior such as 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 behavior near asymptotes, then plotting points to sketch the curve. Following a systematic seven-step process ensures a complete and accurate graph.