Question

What are the holes (if any) in this function: `f( x ) = \frac { x ^ { 2} - 14x + 49} { x ^ { 2} - 10x + 21}`?

Answer 1

This `f(x)` has a hole at `x=7`. It also has a vertical asymptote at `x=3` and horizontal asymptote `y=1`.

Explanation:

We find:

`f(x) = (x^2-14x+49)/(x^2-10x+21)`

`color(white)(f(x)) = (color(red)(cancel(color(black)((x-7))))(x-7))/(color(red)(cancel(color(black)((x-7))))(x-3))`

`color(white)(f(x)) = (x-7)/(x-3)`

Note that when `x=7`, both the numerator and the denominator of the original rational expression are `0`. Since `0/0` is undefined, `f(7)` is undefined.

On the other hand, substituting `x=7` into the simplified expression we get:

`(color(blue)(7)-7)/(color(blue)(7)-3) = 0/4 = 0`

We can deduce that the singularity of `f(x)` at `x=7` is removable - i.e. a hole.

The other value at which the denominator of `f(x)` is `0` is `x=3`. When `x=3` the numerator is `(color(blue)(3)-7) = -4 != 0`. So we get a vertical asymptote at `x=3`.

Another way of writing `(x-7)/(x-3)` is:

`(x-7)/(x-3) = ((x-3)-4)/(x-3) = 1-4/(x-3) -> 1` as `x->+-oo`

So `f(x)` has a horizontal asymptote `y=1`.