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(D) For two functions $f: A \rightarrow B$ and $g: B \rightarrow A$ to be inverses of each other, they must satisfy $g(f(x)) = x$ for all $x \in A$ an

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$A=B=R \setminus R^{-}$

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(D) For two functions $f: A \rightarrow B$ and $g: B \rightarrow A$ to be inverses of each other, they must satisfy $g(f(x)) = x$ for all $x \in A$ and $f(g(x)) = x$ for all $x \in B$.Given $f(x) = x^2$ and $g(x) = x^{1/2}$.For $g(x) = \sqrt{x}$ to be defined, we must have $x \geq 0$ for all $x \in B$. Thus, $B = [0, \infty) = R \setminus R^{-}$.For $f(g(x)) = (x^{1/2})^2 = x$, this holds for all $x \in B$.For $g(f(x)) = (x^2)^{1/2} = |x|$, we require $|x| = x$, which implies $x \geq 0$. Thus, $A = [0, \infty) = R \setminus R^{-}$.Therefore, $f(x)$ and $g(x)$ are inverse functions to each other when $A = B = R \setminus R^{-}$.

Let $f: A \rightarrow B$ and $g: B \rightarrow A$ be defined as $f(x)=x^2 \forall x \in A$ and $g(x)=x^{1/2} \forall x \in B$. $f(x)$ and $g(x)$ are inverse functions to each other when

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