In Histogram Matching or Specification, z = G-1[T(r)], r and z are gray level of input and output image and T & G are transformations, to confirm the single value and monotonous of G-1 what of the following is/are required?

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In Histogram Matching or Specification, z = G<sup>-1</sup>[T(r)], r and z are gray level of input and output image and T & G are transformations, to confirm the single value and monotonous of G<sup>-1</sup> what of the following is/are required?