If $f: A \rightarrow B$ and $g: B \rightarrow C$ are two functions such that $g \circ f: A \rightarrow C$ is…
If $f: A \rightarrow B$ and $g: B \rightarrow C$ are two functions such that $g \circ f: A \rightarrow C$ is a bijection, then which one of the following is always true?
f and g are bijections
f is an injection and g is a surjection
f is a surjection and g is an injection
f is a bijection but g is not a bijection
Solution
If $f: A \rightarrow B$ and $g: B \rightarrow C$ are two function such that $g \circ f: A \rightarrow C$ is bijection.
As, $g \circ f$ is one-one for $x, y$
Let
$f(x)=f(y)$
$g \circ(f(x))=g \circ(f(y))$
$
\Rightarrow \quad g f(x)=g f(y) \Rightarrow x=y
$
$\therefore f$ is one one.
Now, Let $Z \in C$
As gof is onto for every element $Z \in C$ there exist $y \in A$ that is in domain of gof.
$
\begin{gathered}
g \circ f(y)=Z \\
g(f(y))=Z
\end{gathered}
$
$f$ lies in range of $B$ as $f: A \rightarrow B$.
For $\quad a, y \in A, f(y) \in B$, let $f(y)=x$.
For every element $Z \in C$, we have at $X \in B$.
Such that
$
g(x)=Z \text {. }
$
$\therefore g$ is onto
Hence, $f$ is an injection and $g$ is a surjection