Let A be an orthonormal matrix. Let $f$ be a function of which it is known $f(x)=g(x^{T}x)$. Finally, let $\hat{f}=f(A^{-1}x)$. Show $f=\hat{f}$.
My attempt
Knowing A is orthonormal, there exists inverse $A^{-1}$, which is also orthonormal. From this we can deduce $x^Tx=(A^{-1}x)^TA^{^-1}x$. Hence it holds: $$ \hat{f}(x)=f(A^{^-1}x)=g((A^{^-1}x)^TA^{^-1}x)=g(x^Tx)=f(x)$$
Is this correct?
It is incomplete, since you did not say why is it that $(A^{-1}x)^TA^{-1}x=x^Tx$. It is not hard. Since $A$ is orthogonal, $A^{-1}=A^T$ and therefore$$(A^{-1}x)^TA^{-1}x=(A^Tx)^TA^{-1}x=x^TAA^{-1}x=x^Tx.$$