How are discontinuities and zeros determined in rational functions? & what ways can they be accounted for graphically?

Question

arthur326 · Answer

A rational function is a function of the form  $\dfrac{f(x)}{g(x)}$, where $f$ and $g$ are polynomial functions.

The first thing we note is that whenever $g(x) = 0$, there is a discontinuity. (This makes sense: the function is undefined there.)

To study the function further, we cancel all the common factors of $f$ and $g$ to obtain an equivalent function (with the original function's domain) of the form $\dfrac{p(x)}{h(x)}$, where $p$ and $h$ have no common factors.

Now whenever $p(x) = 0$ the function has a zero. Graphically, this means it intersects the $x$-axis at that point. Whenever $h(x) = 0$, the function is undefined and has a vertical asymptote there -- a discontinuity. This happens because as the denominator approaches 0 the fraction grows arbitrarily large. Lastly, at zeros of $g(x)$ which are not zeros of $h(x)$, the function has a removable discontinuity (a "hole"). This is just because the original function is undefined at that point, but we can see that the function does not grow arbitrarily large when we simplify it by cancelling all the common factors.

sushibuns · Answer

Thank you so much! This really helped!

arthur326 · Answer

Great! I'm glad to hear it! :)

likeabossssssss · Answer

$\Huge\color{#EB00FF}{\text{WELCOME}}$ $\Huge\color{blue}{\text{TO}}$ $\Huge\color{green}{\text{OPEN}}$$\Huge\color{purple }{\text{ STUDY!!!!!!!!!!!}}$ $\Huge\heartsuit$