Who can solve floor(y) = x for y?

Question

I am graphing a staircase and am using y = floor(x) for the part of the staircase that someone would put their feet on, and x = floor(y) for the part that connects each part of the stairs, and I would like to solve "floor(y) = x" for y, so I can know what the opposite of floor is. By opposite (since my term may be wrong) I mean in the same way that Sine, Cosine, Tangent, Add, Subtract, Divide, Multiply, and many others have their opposites, (some of which were mentioned in that small list) and I would like to know what the opposite to floor is. I have searched all over for this answer but cannot find anything... Might I be the first to ponder this? I really hope not...

Answer 1

The sense of "opposite" that I think you're looking for is "inverse", as in "the inverse of a function". The floor function doesn't have a proper inverse because it is not a "one-to-one function". Let me put that in more concrete terms:

We know that floor(2.1)=2, and also floor(2.5)=2. Thus, if you want to solve "floor(y)=2" for y, there's no way to say whether the answer should be 2.1, or 2.5, or any of the infinitely many other numbers that are $\geq$ 2 and <3.

The best you can do is say that the "solution" to floor(y)=x is the entire interval: $[x,x+1)$.

Does this help?

Answer 2

If $x \notin \mathbb{Z}$, there is no solution.

If $x \in \mathbb{Z}$, we can only conclude that $x \leq y < x+1$.

Answer 3

The floor function doesn't have an inverse, so in order to graph your staircase, you would need to put your equations in parametric mode. If your calculator allows parametric mode, here's the equations you would enter:

$x_1=t$, $y_1=floor(t)$

$x_2=floor(t)+1$, $y_2=t$