How I found the value

The first option is always Solve (also NSolve). As @Jason showed, using Solve with a specific value will give you all possible solutions

f[x_] = x^2
Solve[f[x] == 1.2, x]

{{x -> -1.09545}, {x -> 1.09545}}

On the other hand as @Mark showed,

Solve[y == f[x], x]

{{x -> -Sqrt[y]}, {x -> Sqrt[y]}}

will give you all possible analytical solutions which you can use to generate values of x for any y. However these methods can be tricky when you have large and complicated functions (specially trigonometric functions where you might have to specify the domain like 0<x<2Pi as well). In that case you can use FindRoot

FindRoot[f[x] == 1.2, {x, 1.0}]
FindRoot[f[x] == 1.2, {x, -1.0}]

{x -> 1.09545} {x -> -1.09545}

FindRoot requires an initial guess and gives only one numerical solution closest to the initial guess.