are the functions also $1-1$ and surjective Knowing that $f: X \to Y \text{ and } g:Y \to Z$ are $1-1$ and surjective,are then also $f^{-1}: Y \to X$ and $g \circ f:X \to Z$ $1-1$ and surjective or just $1-1$ ?

1) $f^{-1}$ is surjective and 1-1, because for every $x \in X   \ \ f^{-1}(f(x)) = x$ and if $f^{-1}(a) = f^{-1}(b)$ then $a = f(f^{-1}(a)) = f(f^{-1}(b)) = b$
2) $g \circ f $ is surjective and $1-1$, because if $g \circ f (a) = g \circ f (b)  $ then $f(a) = f(b) $ and so $a = b$. Moreover for every $z \in Z $,  let $b \in Y $ be such that $g(b) = z $ and $a \in X $ such that $f(a) = b $. Then $g \circ f (a) = z$
