# Let f(x) = (x)/(x^2-4) (a) State the y-intercept. (b) State the vertical asymptote(s). (c) State...

## Question:

Let {eq}f(x) = \frac{x}{x^{2}-4} {/eq}

(a) State the y-intercept.

(b) State the vertical asymptote(s).

(c) State the horizontal asymptote.

## Rational Function:

Rational functions have characteristics that differentiate them from other types of functions. The first is that rational functions have asymptotes, which are straight horizontal or vertical lines. Another characteristic is the restrictions that it has in its domain depending on the polynomial expression that is in the denominator

Function: {eq}f\left(x\right)=\frac{x}{x^2-4} {/eq}

Part a: State the y-intercept.

The y-intercept is found when the value of x is zero.

{eq}\begin{align*} f\left(x\right)&=\frac{x}{x^2-4} \\ y&=\frac{x}{x^2-4} \\ y&=\frac{\left(0\right)}{\left(0\right)^2-4} \\ y&=0 \end{align*} {/eq}

Part b: State the vertical asymptotes.

The domain is given by the numbers that make zero the denominator.

{eq}\begin{align*} x^2-4&=0 \\ \left(x-2\right)\left(x+2\right)&=0 \\ x-2&=0 \quad x=2 \\ x+2&=0 \quad x=-2 \end{align*} {/eq}

The domain is: {eq}\:\left(-\infty \:,\:-2\right)\cup \left(-2,\:2\right)\cup \left(2,\:\infty \:\right) {/eq}

The numbers that make the denominator zero are the vertical asymptotes. The vertical asymptotes are: {eq}x=-2 \quad x=2 {/eq}

Part c: State the horizontal asymptote.

If the degree of the denominator is greater than the degree of the numerator, the horizontal asymptote is the x axis, that is, y = 0. In this case, the degree of the denominator is 2 while the degree of the numerator is 1. The horizontal asymptote is: {eq}y=0 {/eq}