Answer
Yes, if we know the antiderivative $F(x)$, we know the function $f$, provided that the antiderivative $F(x)$ is differentiable.
Work Step by Step
If $F(x)$ is an antiderivative of $f(x)$, that is $ F'(x)=f(x)$, then the indefinite integral of $f$ is
$\displaystyle \int f(x)dx=F(x)+C.$
If we know a particular antiderivative $g(x)=F(x)+C$,
then our function is the derivative of $g$
$f(x)=g'(x)$
$=\displaystyle \frac{d}{dx}[F(x)+C]$
$=F'(x)+0$
$=F'(x)$
So, yes, if we know the antiderivative $F(x)$, we know the function $f$, provided that the antiderivative $F(x)$ is differentiable.
