BNL lunch break.
Adam was talking about a function T(f(x),g(x)) that takes any linear function f(x) and apply g(x), but make it seems like f(x) is the x-axis.
Without drawing a graph, it's quite ambiguous of what the above sentence mean.
A graph example, where f(x) = x, g(x) = sin(x).

So I told him, to do that, just do the g(x) on x-axis and then rotate the entire graph with the origin as center until the original x-axis is on f(x).

He then asked, what if f(x) is not a linear function.

For a second I can't think of something out, and soon, I was distracted so I said I will look up tonight.

On the road back to my office, the answer popped up. My answer is trivial and it's just one way to see the question, and I believe there must be other ways.

T(f(x),g(x)) generates a graph, where each point of the graph is the point g(x) away from (x, f(x)) and on the line that perpendicular to f'(x) and intersects (x,f(x)).

Finding a universal equation for this T(f(x),g(x)) to graph it:
It's trivial. The above description almost instantly lead to the result.

I have to use parametric, because can be a one to many function even if both functions are one to one functions.
If f(x) is not differentiable at some points, then the graph is not defined at those points.
Some nice graphs generated from Mathematica
f(x) = x^2, g(x) = floor(|x|)

f(x) = sin(x), g(x) = sin(4x)

Ones I used maxima + gnuplot
f(x) = sin(x), g(x) = floor(x)

f(x) = sin(x), g(x) = x

f(x) = sin(x), g(x) = log(x)

f(x) = sin(x), g(x) = x^2

f(x) = sin(x), g(x) = sqrt(2)

I hope I know what this stuff Adam thought of is called.