Question

Follow the seven steps to graph each rational function. f(x)=− 1/(x2−4)

Accepted Answer

Identify the domain of the function by finding the values of $$x$$ that make the denominator zero. Set the denominator equal to zero: $$x^{2} - 4 = 0$. Solve the equation x$$^{2}$$ - 4 = 0 by $$factoring it as $$(x - 2)(x + 2) = 0$$, which gives the values $$x = 2$$ and $$x = -2. $$These values are excluded from the domain because they make the denominator zero, causing vertical asymptotes. Determine the vertical asymptotes by noting that the function is undefined at $$x = 2$$ and $$x = -2. So$$, draw vertical dashed lines at these $$x-$$values. Find the horizontal asymptote by analyzing the degrees of the numerator and denominator. The numerator is a constant ($$-1$$), and the denominator is a quadratic ($$x^{2} - 4$). Since the degree of the denominator is greater than the numerator, the horizontal asymptote is y = 0$. Create a table of values by choosing x$-values around the vertical asymptotes (for example, values less than -2$, between -2$ and 2$, and greater than 2$), then calculate the corresponding f(x$$)$$ values to understand the behavior of the graph in each interval.

Follow the seven steps to graph each rational function. f(x)=− 1/(x²−4)

Concetti chiave

Rational Functions

Asymptotes of Rational Functions

Graphing Steps for Rational Functions