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# If $f(x)=\sqrt x (x \geq 0)\: and \: g(x) = x^2-1$ are two real functions, find $fog\: and \: gof.\: Is\: fog = gof.$

Toolbox:
• Given two functions $f:A \to B$ and $g:B \to C$, then composition of $f$ and $g$, $gof:A \to C$ by $gof (x)=g(f(x))\;for\; all \;x \in A$
• Given two functions $g:A \to B$ and $f:B \to C$, then composition of $g$ and $g$, $fog:A \to C$ by $fog (x)=f(g(x))\;for\; all \;x \in A$
Given :
$f(x)=\sqrt x,x\geq 0$
$g(x)=x^2-1$
$fog=\sqrt{x^2-1}$
$gof=(\sqrt x)^2-1$
$\qquad=x^2-1$
$\therefore fog\neq gof$