# What is the domain and range, x- and y-intercept, and horizontal and vertical asymptote of y =...

## Question:

What is the domain and range, x- and y-intercept, and horizontal and vertical asymptote of {eq}y = \frac{400 + 40x}{x}? {/eq}

## Rational Functions:

Rational functions are functions that are a quotient of two polynomials. The domain is all real numbers except whatever number makes the denominator zero. The x-intercept can be found by setting y=0. The y-intercept can be found by setting x=0. For horizontal asymptotes, we look at the degrees of the numerator and the denominator. If the degree of the numerator is smaller than or equal to the degree of the denominator, then there is a horizontal asymptote. THe vertical asymptotes exist wherever the denominator is zero.

Observe that we can rewrite y as:

{eq}y = \dfrac {400}x +40 {/eq}.

Now the domain is all numbers except 0, as the denominator is zero when x=0.

The range is all real numbers, as it graphs a hyperbola.

The x-intercept:

{eq}0 = \frac{400+40x}x \\ 0 = 400+40x \\ -400 = 40 x \\ -10 = x {/eq}

The y-intercept does not exist since when x=0 the y value is not defined.

Horizontal asymptote is at y=0 since {eq}y= \dfrac{400}x {/eq} has a higher degree in the denominator than the numerator.

The vertical asymptote is at x=0, since that is when the denominator of y is 0.