Boundedness of an expression

Question

First post on this part of the stackexchange network, I was just wondering if Mathematica allows one to quickly see if an expression is always within a certain range given values for the parameters it depends on?

For example, I have an expression $X = X(a,b,c,d,e,f)$ and know that two of those six variables are always positive and the other four are always negative. I am finding it difficult to see in general whether this expression is within a certain range (e.g between $-1$ and $1$) by hand so was wondering whether this software would do it for me? It is part of a physics project and I have tried some numerical values by hand and the expression is indeed within the range required however seeing the general case is always more satisfactory.

Answer 1

There's no single BoundedQ function that I know of, but there are several related functions:

FunctionRange tries to compute the range of a function:

range = FunctionRange[1/x, x, y]
(* y < 0 || y > 0 *)

It is clear that this is not bounded, but we can also try to get that result programmatically:

BoundedRegionQ@ImplicitRegion[range, y]
(* False *)

Or like this:

range /. {{y -> Infinity}, {y -> -Infinity}}

We see that both Infinity and -Infinity satisfy the inequalities, so the range is not bounded.

FunctionRange can also take conditions:

FunctionRange[{1/x, x > 1}, x, y]
(* 0 < y < 1 *)

With this condition it is bounded.

FunctionRange is meant to compute the exact range I can imagine that in many situations it will be much easier to prove that a function is or isn't bounded than to compute the actual bounds. In those cases this might be useful:

upperBoundGuess = 10

Resolve@Exists[x, 1/x > upperBoundGuess]
(* True *)

Resolve[Exists[x, Sin[x] > upperBoundGuess], Reals]
(* False *)

It tells us that 1/x can get larger than 10 but Sin[x] cannot. For the second case we needed to specify that we're working over the set of reals as Sin[Pi/2 + 3 I] is indeed real and greater than 10.

You can also use Reduce in place of Resolve.