Binary Operations and Inverse elements in groups and functions

Question

Suppose we have group $G$ in relation to binary operation $*$ and $a\in G$ and function $f$ from $G$ to $G$ ($f:G\to G$), and for each $x\in G$ $f(x)=a^{-1} * x * a$.

Prove that if $b,c\in G$ are inverse elements to each other, then $f(b)$ and $f(c)$ are inverse elements to each other as well.

My way of answer: $b$ and $c$ are inverse elements, meaning that $f(b)=a^{-1} * b * a$ is the inverse of $f(c)=a^{-1} * c * a$, but I am not quite sure.

Answer 1

Just calculate $f(b)^{-1}$ by folowing the group laws: \begin{align}f(b)^{-1}&= (a^{-1}ba)^{-1} \\&= a^{-1}b^{-1}(a^{-1})^{-1} \\&=a^{-1}ca \\&= f(c)\end{align}

Answer 2

We have

$$\begin{align} f(b)f(c)&=(a^{-1}ba)(a^{-1}ca)\\ &=a^{-1}b(aa^{-1})ca\\ &=a^{-1}(bc)a\\ &=a^{-1}ea\\ &=e, \end{align}$$

which implies $f(b)=f(c)^{-1}$ by uniqueness of inverses.