Graph each rational function. ƒ(x)=5x/(x²-1)

Verified step by step guidance

1. Identify the rational function given: \(f(x) = \frac{5x}{x^{2} - 1}\). Notice that the denominator can be factored.
2. Factor the denominator: \(x^{2} - 1 = (x - 1)(x + 1)\). This helps us find the domain restrictions and vertical asymptotes.
3. Determine the domain by setting the denominator not equal to zero: solve \(x^{2} - 1 \neq 0\), which means \(x \neq 1\) and \(x \neq -1\). These values are vertical asymptotes.
4. Find the horizontal asymptote by comparing the degrees of the numerator and denominator. Since the degree of the denominator (2) is greater than the numerator (1), the horizontal asymptote is \(y = 0\).
5. Find the x-intercept by setting the numerator equal to zero: \$5x = 0\(, so \)x = 0\(. Find the y-intercept by evaluating \)f(0)\(, which is \)\frac{0}{0^{2} - 1} = 0$. Use these points and asymptotes to sketch the graph.

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, zeros, and behavior of these functions is essential for graphing, as they often have asymptotes and discontinuities where the denominator is zero.

Vertical and Horizontal Asymptotes

Vertical asymptotes occur where the denominator equals zero and the function is undefined, indicating values the graph approaches but never touches. Horizontal asymptotes describe the end behavior of the function as x approaches infinity or negative infinity, determined by comparing the degrees of numerator and denominator polynomials.

Graphing Techniques for Rational Functions

Graphing involves finding intercepts, asymptotes, and analyzing the function's behavior near discontinuities. Plotting key points and understanding symmetry or sign changes help create an accurate sketch of the function's curve.