From the definition of rational function $f(x) = \frac{p(x)}{q(x)}$ , where P(x) and q(x) are polynomials and q(x) ≠ 0
so for the function $f(x) = \frac{x^{-2}+3 }{x-5}$ by the definition f(x) isnt rational since the numerator is not polynomial
but by multiplying both numerator and denominator by $x^{2}$ we get
$f(x) = \frac{3x^{2}+1 }{x^{3}-5x^{2}}$ which is rational
and can we say that both functions are equal at every point?
They are not equal at every point. They are equal where they are both defined, on the intersection of their domains, which is all real numbers except $0$ and $5$. (So they are not equal at $0$ or at $5$ since they aren't even defined there.)