Graph each rational function. See Examples 5–9.
$\text{ƒ}\left(x\right)=x/(4-x^2)$

Verified step by step guidance

1. Identify the rational function given: \(f(x) = \frac{x}{4 - x^2}\).
2. Determine the domain by finding values of \(x\) that make the denominator zero. Solve \$4 - x^2 = 0$ to find vertical asymptotes.
3. Find the vertical asymptotes by setting the denominator equal to zero and solving for \(x\): \$4 - x^2 = 0\( implies \)x^2 = 4$, so \(x = \pm 2\).
4. Find the horizontal asymptote by analyzing 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 intercepts: For the \(y\)-intercept, evaluate \(f(0)\); for the \(x\)-intercept, set the numerator equal to zero and solve for \(x\). Then, plot these points along with the 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). Understanding its domain, zeros, and behavior depends on analyzing both numerator and denominator. For example, f(x) = x/(4 - x^2) involves a quadratic denominator that affects the function's properties.

Domain and Vertical Asymptotes

The domain of a rational function excludes values that make the denominator zero. These values often correspond to vertical asymptotes, where the function approaches infinity or negative infinity. For f(x) = x/(4 - x^2), setting 4 - x^2 = 0 finds vertical asymptotes at x = ±2.

Graphing Rational Functions and Asymptotes

Graphing involves plotting intercepts, identifying asymptotes, and analyzing end behavior. Horizontal or oblique asymptotes describe the function's behavior as x approaches infinity. For f(x) = x/(4 - x^2), the horizontal asymptote is y = 0, since the degree of the denominator is higher than the numerator.