Say we want to determine the range of the transformation A x. And we call the solution vectors b. So if we want to determine all possible b's, are we going to have to put A into reduce row echelon form? or is there another way?
Or alternatively, given a transformation A and a b, is there a way to tell whether b is in the range without reducing Ax=b into reduced row echelon form?
You can check the rank of $$\left[ \begin{array}{c|c}A &b \end{array} \right]$$ if that is an alternative way that you are looking for (note that row reduced echelon form is also a way of to compute the rank).
In particular, if $A$ is fat i.e. number of rows $m$ is less than the number of columns $n$, then infinitely many solution exists. If $m=n$ then either the matrix is invertible and every $b$ has a solution in the form of $x=A^{-1}b$ or $A$ is singular and you can reduce it to either a fat or tall matrix with full row/column rank.
The last option is the tall matrix case. Then you can simply check if $$\operatorname{rank}\left[ \begin{array}{c|c}A &b \end{array} \right]>\operatorname{rank}A$$ In case of a positive answer : No $x$ can lead to that particular $b$ (which is usually the case). Hence the popularity of least squares solution.