The formula \(\displaystyle{T}={f{{\left({x}\right)}}}={x}^{{{\frac{{{3}}}{{{2}}}}}}\) which converts planet orbital radius to orbital period. If \(\displaystyle{T}={f{{\left({x}\right)}}}\) and f(x) is a bijective mapping (both one-to-one and surjective) then its inverse exists and thus \(\displaystyle{x}={{f}^{{-{1}}}{\left({T}\right)}}\).

Calculation:

a) Consider the given formula \(\displaystyle{T}={f{{\left({x}\right)}}}={x}^{{{\frac{{{3}}}{{{2}}}}}}\) with \(\displaystyle{x}\geq{0}\), which is an increasing function satisfying bijectivity. Now, raising both the sides to power \(\displaystyle{\frac{{{2}}}{{{3}}}}\) we get \(\displaystyle{T}^{{{\frac{{{2}}}{{{3}}}}}}={\left({x}^{{{\frac{{{3}}}{{{2}}}}}}\right)}^{{{\frac{{{2}}}{{{3}}}}}}\). This gives, \(\displaystyle{x}={{f}^{{-{1}}}{\left({T}\right)}}={T}^{{{\frac{{{2}}}{{{3}}}}}}\)

b) From part (a) we have seen \(\displaystyle{x}={{f}^{{-{1}}}{\left({T}\right)}}={T}^{{{\frac{{{2}}}{{{3}}}}}}\). Thus inverse of T calculates orbit radius as a function of orbital period.

Conclusion:

Thus, we found out that \(\displaystyle{x}={{f}^{{-{1}}}{\left({T}\right)}}={T}^{{{\frac{{{2}}}{{{3}}}}}}\) is the inverse of T and it calculates orbit radius as a function of orbital period. We can also explain it in another way. If planet orbital period function is represented by \(\displaystyle{\left({x}\right)}={x}^{{{\frac{{{3}}}{{{2}}}}}}\), its inverse can be expressed as \(\displaystyle{T}^{{-{1}}}{\left({x}\right)}={x}^{{{\frac{{{2}}}{{{3}}}}}}\). For T(x), the argument x represents the number of times the planet with orbital period T is farther from sun than Earth is.