Question

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

Answer 1

Given, f : R⁺ → R⁺ and g : R⁺ → R⁺
So, fog : R⁺ → R⁺ and gof : R⁺ → R⁺
Domains of fog and gof are the same.
$$\begin{gathered} \left(fog\right)\left(x\right)=f\left(g\left(x\right)\right)=f\left(\sqrt{x}\right)={\left(\sqrt{x}\right)}^2=x \\ \left(gof\right)\left(x\right)=g\left(f\left(x\right)\right)=g\left(x^2\right)=\sqrt{x^2}=x \\ \text{So,}\left(fog\right)\left(x\right)=\left(gof\right)\left(x\right),\forall x\in R^{+} \end{gathered}$$
Hence, fog = gof