Get solutions of an equation given its output Lets assume I have a function $f(x_1, x_2, x_3)$ that takes $x_1, x_2, x_3$ as inputs. For the sake of simplicity, lets assume that this $f(x)$ is a linear equation. 

My question is- if I decide that the output of my function has to be an arbitrary value- $v$, can I generate all or a fixed number of the possible solutions or combinations of values that when passed to $f$ will generate $v$?

If yes, how do I achieve this?
Since you assume that $f(x)$ is linear, it takes the form $a_0 + a_1x_1+a_2x_2+a_3x_3$. Let $v'=v-a_0.$ If $\mathbf{a}=(a_1,a_2,a_3)$, you have to find all $\mathbf{x}\in\mathbb{R}^3$ such that $\mathbf{a}.\mathbf{x}=v'$.

To find $\mathbf{x}$, we can do a coordinate transformation so that $\mathbf{a}$ is aligned with the $x$-axis and then $\mathbf{x}$ is any vector whose $x$ component equals $v'$ in the new system. Use inverse coordinate transformation to find $\mathbf{x}$ in the original coordinate system.

I'm sorry for the partial answer but I've forgotten how to do these steps. Maybe someone else can elaborate.
Yes, for a linear function it is always possible to reconstruct $x$ from $v$ given that $v$ is a solution.

In general, for $f:\mathbb{R^n}\to\mathbb{R^m}$ to do that we simply need to solve the linear system $Ax=v$, where $A$ is the matrix representation of $f$ for a given basis.

For the particular case of $f:\mathbb{R^3}\to\mathbb{R}$, we have that $x_1$,$x_2$ and $x_3$ scalars and we need to solve

$$a_1x_1+a_2x_2+a_3x_3=v$$

which has infinitely many solutions, that is setting $x_2=s$ and $x_3=t$

$$x_1=\frac{v-a_2s-a_3 t}{a_1}$$