I have a function which maps a vector to a scalar.
$$ f(q, r) = 135 + q + \frac {r( 39 - |r| )}{2} $$
The domain for q is: $$ q \in \mathbb{Z}, -9 <= q <= 9 $$
The domain for r is: $$ r \in \mathbb{Z}, -9 <= r <= 9 $$
The range of this function (u) is : $$ u \in \mathbb{N}, 0 <= u <= 270 $$
I want to create the function
$$f^{-1}(x,y) $$
But I am unable to.
I've tried breaking the problem into pieces to get try to get a better handle on it:
but I can't solve the simpler problem of inversing mapping the r component of the vector either.
$$ f(x) = 135 + \frac {x( 39 - |x| )}{2} $$
To create the function inverse function
$$ f^{-1}(x) $$
My first thought was the f(x) could be a quadratic (and I could create the inverse function by completing the square) but after graphing it is nothing like a quadratic.
