Let R+ be the set of all non-negative real numbers.


Let $R^{+}$be the set of all non-negative real numbers. If $f: R^{+} \rightarrow R^{+}$and $g: R^{+} \rightarrow R^{+}$are defined as $f(x)=x^{2}$ and $g(x)=+\sqrt{x}$, find fog and gof. Are they equal functions?


Given, $f: R^{+} \rightarrow R^{+}$and $g: R^{+} \rightarrow R^{+}$

So, fog: $R^{+} \rightarrow R^{+}$and gof: $R^{+} \rightarrow R^{+}$

Domains of fog  and gof  are the same

$(f o g)(x)=f(g(x))=f(\sqrt{x})=(\sqrt{x})^{2}=x$

$(g o f)(x)=g(f(x))=g\left(x^{2}\right)=\sqrt{x^{2}}=x$

So, $(f o g)(x)=(g o f)(x), \forall x \in R^{+}$

Hence, fog gof


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