Graph each rational function. See Examples 5–9.ƒ(x)=x/(4−x2)ƒ(x)=$$x/(4-x^2)$$

Ch. 3 - Polynomial and Rational Functions

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Ch. 3 - Polynomial and Rational Functions

문제 70

4장, 문제 70

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

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Identify the rational function given: $f(x) = \frac{x}{4 - x^2}$.

Determine the domain by finding values of $x$ that make the denominator zero. Solve $4 - x^2 = 0$ to find vertical asymptotes.

Find the vertical asymptotes by setting the denominator equal to zero and solving for $x$: $4 - x^2 = 0$ implies $x^2 = 4$, so $x = \pm 2$.

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 $y = 0$.

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.

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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.

Graph each rational function. See Examples 5–9.ƒ(x)=x/(4−x2)ƒ(x)=x/(4-x^2)