Let $Y =\left\{n^{2}: n \in N \right\} \subset N .$ Consider $f: N \rightarrow Y$ as $f(n)=n^{2}$ Show that $f$ is invertible. Find the inverse of $f$
Let $Y =\left\{n^{2}: n \in N \right\} \subset N .$ Consider $f: N \rightarrow Y$ as $f(n)=n^{2}$ Show that $f$ is invertible. Find the inverse of $f$
An arbitrary element $y$ in $Y$ is of the form $n^{2}$, for some $n \in N .$ This implies that $n=\sqrt{y} .$ This gives a function $g: Y \rightarrow N$, defined by $g(y)=\sqrt{y} .$ Now, $gof\,(n)$ $=g\left(n^{2}\right)$ $=\sqrt{n^{2}}=n$ and $fog (y)=f(\sqrt{y})=$ $(\sqrt{y})^{2}=y,$ which shows that $g o f=I_{N}$ and $f o g=I_{Y} .$ Hence, $f$ is invertible with $f^{-1}=g$
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- [IIT 1999]
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