When do we call two functions equal? For example, are $f(x)=x$ and $g(x)=x^{2}/x$ equal? We should note that $f(0)=0$, but because division by zero is not allowed, $g(0)$ does not exist. In other words, $0\in Dom(f)$ but $0\not\in Dom(g)$. The graphs of $f$ and $g$ are shown in Figure 1. As we can see the point $(0,0)$ has been excluded from the graph of $y=g(x).$

Two functions $f$ and $g$ are equal (or identical) if and only if

- $f$ and $g$ have the same domain, i.e. $Dom(f)=Dom(g)$,
- $f(x)=g(x)$ for every $x$ in their domains

If two functions $f$ and $g$ are equal, then

- the graphs of $f$ and $g$ are identical
- $f$ and $g$ have the same range, i.e. $Rng(f)=Rng(g)$.