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In Mathematica 8.0.1.0 on 32-bit Linux, the expression
InverseFunction[0 &]@0
returns 33/10. (The same occurs for other integer and rational values; I'm using 0 as an example.)
According to the documentation for InverseFunction:
As discussed in Functions That Do Not Have Unique Values, many mathematical functions do not have unique inverses. In such cases, InverseFunction[f] can represent only one of the possible inverses for f.
As a constant function 0& will return 0 regardless of its input, it has infinitely many inverse functions (each of which is defined only at 0). So as defined, this answer is within the specification.
The mystery is, why does it give 33/10 rather than any other value?
481
Zero doesn't have a multiplicative inverse as multiplicative inverse is the reciprocal for a number “a”, denoted by 1/a, is a number which when multiplied by “a” yields the multiplicative identity 1. The product of any real number with zero is zero.
If any horizontal line intersects the graph of f more than once, then f does not have an inverse. If no horizontal line intersects the graph of f more than once, then f does have an inverse. The property of having an inverse is very important in mathematics, and it has a name.
In the real numbers, zero does not have a reciprocal because no real number multiplied by 0 produces 1 (the product of any number with zero is zero).
Not all functions have inverse functions. The graph of inverse functions are reflections over the line y = x. This means that each x-value must be matched to one and only one y-value. Functions that meet this criteria are called one-to one functions.
That number appears in a number of instances. Take for instance:
FindInstance[x == x, x, Reals]
{{x->33/10}}
I've seen discussions of this number come up before. It's basically just some result of how Mathematica is implemented. You'll get this sometimes when you ask Mathematica to do something that boils down to "Pick a Random Real number". It doesn't have any real special meaning.
123
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