PRE-CALCULUS

Let’s begin discussing the graphs of a rational function which can have discontinuities. A rational function has a polynomial in the denominator which means you’re dividing something that’s very big quantity. So you can be sure with the bottom of the fraction it has.

It’s impossible that some values of x is divided by zero. If it shows like that, the value is off limits.

Let f(x)=(x+2)/(x-1) and when x = 1 the function value becomes f(1)=(1+2)/(1-1)=3/0. In this case 0 is the denominator.
From this function, choosing x=1 is a bad idea. It makes a break in function graph.
For example : Suppose we have a graph of f(x)=(x+2)/(x-1). Insert 0 for x, so f(0)=(0+2)/(0-1)=-2. So the point of function in graph is (0,-2).
Now, next to try insert x=1. So we get f(1)=(1+2)/(1-1)=3/0. We know that is impossible. It also means that the graph of function we make has no any point of y when x=1. We can also call the graph is break or discontinuity.
Rational functions don’t always work this way. Take a graph of f(x)=1/(x+1), so f(1)=1/(1+1)=1/2.
Not all of rational functions will give zero in denominator, so the graph is smooth or not broken.
But don’t forget! The denominator of rational function can be zero!