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# If functions $f\;:\;A \rightarrow B\;and\;g\;:\;B \rightarrow A$ satisfy g o f =$I_A$ ,then is f one-one and g onto

Toolbox:
• A function $f:X \rightarrow Y$ where for every $x1, x2 \in X, f(x1) = f(x2) \Rightarrow x1 = x2$ is called a one-one or injective function
• A function$f : X \rightarrow Y$ is said to be onto or surjective, if every element of Y is the image of some element of X under f, i.e., for every $y \in Y$, there exists an element x in X such that $f(x) = y$.
Given $\; gof=I_A$, $f:A \to B$, $g: B \to A$.
Step1: Injective or One-One function:
$gof(x)=x \qquad g:B \to A$
$g(f(x))=x \rightarrow g(f(x_1))=x_1\;and \;g(f(x_2))=x_2$
$f(x_1)=f(x_2)=>x_1=x_2$
Hence f must be one -one
Step 2: Surjective or On-to function:
$gof(x)=x$
$g(f(x))=x$
=>for every element $f(x) \in A$ then exists an element $x \in B$ such that
$g(f(x))=x$
=>Hence g is onto
Solution:Thus function f is one-one and g is onto
edited Mar 30, 2013