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What are some examples of triplet of functions- $(f,g,h)$ such that:

$f(h(a,b,c),c)=a$, and $g(h(a,b,c),c)=b$

The functions $f, g, h$ are defined for all $a, b, c, d ∈ R$

Hint: We observe such triplet of functions $(f, g, h)$ in the Division operation(for a particular case where $a, b, c, d$ all are positive integers, and $c>b$):

$h (a, b, c) = a×c + b$

$G.I.F.[h(a, b, c)/c] = a$

$mod[h(a, b, c), c] = b$

Here, a= Quotient, b= Remainder, c= Divisor, and h(a, b, c)= Dividend

Basically, in the function "$h$" we have used three parameters- a, b, and c to retrieve an output. But, in the other two functions "$f$" and "$g$", we have used that output and one of the inputs of function "$h$"- c to get the other two inputs- a and b.

So, what are some other triplets of the functions- (f, g, h), apart from the case shown as an example that satisfy the same set of conditions?

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    Do you want continuity? Otherwise you can unpack decimal expansions to do what you want. – 2017-01-17
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    It doesn't matter whether it's continuous or not.....and would you plz elaborate what you've said regarding "unpacking decimal expansion"? – 2017-01-17
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    Consider a two numbers $a,b$ in $(0,1)$. Expand $a$ and $b$ in a unique way (ie no infinite sequence of $9$s) so that $a=0.a_1a_2a_3...$ $b=0.b_1b_2...$, define your function that "zips the two numbers together" so that $h(a,b)=0.a_1b_1a_2b_2a_3b_3....$ and $f$ and $g$ are chosen so that they unpack this construction to get either $a$ or $b$ back. – 2017-01-17

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