Given the problem:
Given $f (x) = x - 3 [2 x] + 2 [3 x]$ is invertible and $f^{- 1} (x) = a x + b [2 x] + c [3 x]$ with a, b and c are integers, find the appropriate matches based on the given solution.
Where [ ] represents greatest integer of x. and $| x |$ is absolute value or modulus of $x$ .
Match the items in Column I with the appropriate expressions or properties in Column II.

Column I

Column II

1 $a + b + c$ A
2
2 $| a + c |$ B
4
3 $| a + b |$ C
1
4 $[\frac{b}{a}]$ D
3
[ ] represents greatest integer of x

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1-A,2-B,3-C,4-D

1-A,2-A,3-B,4-B

1-B,2-B,3-B,4-A

1-A,2-B,3-A,4-B

answer is A.

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Detailed Solution

$\begin{array}{r} \begin{array}{l} y = x - 3 [2 x] + 2 [3 x] \\ I n t e r c h a n g e x a n d y \end{array} \\ x = y - 3 [2 y] + 2 [3 y] \\ y = x - 3 [2 y] + 2 [3 y] \\ 2 y = 2 x - 5 [2 y] + 4 [3 y] \\ [2 y] = [2 x] - 6 [2 y] + 4 [3 y] - (2) \\ x = y - 3 ([2 x] - 6 [2 y] + 4 [3 y]) + 2 [3 y] \\ x = y - 3 [2 x] + 18 [2 y] - 10 [3 y] \\ [3 y = [3 x] - 9 [2 y] + 6 [3 y] \\ 5 = x = 2 y - 3 [2 x] + 2 [3 x] - 3 [2 y] + 2 [3] \\ x = y - 3 [2 x] + 2 [3 x] \\ x = y + 3 [2 x] - 2 [3 x] \end{array}$