What is the maximal domain of these functions?

Question

Need help figuring out, how to find the maximal domains of functions

$f(x) = \frac{x^3-x}{2x^2+1}$

$f(x) = \frac{x^2-5x+6}{x-2}$

Answer 1

I will suppose $f(x)$ should be $f(y)$ and $f(z)$ and that we are working in $\mathbb{R}$.

1. This is a rational expression. The only restriction is that the denominator is not $0$. Since $$2y^2+1=0$$ has no solution then the denominator does not impose any restriction on the domain. Hence $dom(f)=\mathbb{R}$.

2. Same thinking leads to $$z-2\ne 0 \Leftrightarrow z\ne 2.$$ The domain is $\mathbb{R}\backslash\{2\}$.

Answer 2

The second function does not contain $z=2$ in the domain, but it can be extended continuously at this point, thus defined continuous, so the maximal domain would be all real numbers.