# How do you find the x- and y-intercepts of f(x) = (4 - x) / x, then determine whether the graph of f touches or crosses the x-axis at the x-intercept?

Nov 2, 2016

Let's start by determining the $y$ -intercept.

$y = \frac{4 - 0}{0} = \frac{4}{0} = \emptyset$

There is no y-intercept, since there is an asymptote at $x = 0$.

For the x-intercept:

$0 = \frac{4 - x}{x}$

$0 = 4 - x$

$x = 4$

To determine the behaviour of the graph at the $x$-intercept, we need to determine the equation of the horizontal asymptote.

Since the degree of the numerator equals the degree of the denominator, the horizontal asymptote will occur at the ratio between the coefficients of the highest degree in the numerator and denominator.

This would be $- \frac{1}{1} = - 1$.

So, the horizontal asymptote is at $y = - 1$. We can hence conclude that the graph will continue to converge upon this line, but never touch it. Therefore, it will cross the x axis at $x = 4$ and continue going downwards.

Hopefully this helps!