function composition proof with identity functions

Question

I need to prove: $(g \circ f)^{-1}=f^{-1} \circ g^{-1}$.

Is this okay?:

$(g \circ f) \circ (f^{-1} \circ g^{-1})=g \circ (f \circ f^{-1}) \circ g^{-1}=g\circ id_{A} \circ g^{-1}=g \circ g^{-1}= id_{A}$ .