Question

What are the `x`-intercepts of the graph of `y=(x-4)/(x^2+4)`?

Answer 1

`x=+4` is the only zero of `y` and hence the only `x-`intercept

Explanation:

The `x-`intercepts are the zeros of `y` i.e. value(s) where `y=0`

`:. (x-4)/(x^2+4) =0`

Clearly, `x=+4` satisfies the above equation.

The question then arises as to whether or not `y` has any other zeros.

First let's consider `y :x<+4`

In this interval `y<0` since `(x-4)<0` and `(x^2>0)`

`:. y` has no zeros in the interval `x=(-oo, +4)`

Now consider `y:x>+4`

In this interval `y>0` since `(x-4)>0` and `(x^2>0)`

`:. y` has no zeros in the interval `x=(+4, +oo)`

Hence, `x=+4` is the only zero of `y` and hence the only `x-`intercept

This can been visualised by the graph of `y` below.

graph{(x-4)/(x^2+4) [-8.89, 8.89, -4.45, 4.44]}