Let f: A→B, g:B→A be defined as fx=x2∀x∈A and gx=x1/2∀x∈B. fx and gx are inverse functions to each other when

TS EAMCET · Maths · Functions

Let $f : A \to B, g : B \to A$ 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

A $A = B = R$
B $A = R \ R^{-}; B = R \ R^{+}$
C $A = R; B = R \ R^{-}$
D $A = B = R \ R^{-}$

Verified Solution

Answer & Solution

Correct Answer

(D) $A = B = R \ R^{-}$

Step-by-step Solution

Detailed explanation

Let f: A→B, g:B→A be defined as fx=x2∀x∈A and gx=x1/2∀x∈B. Given, fx=x2 ∀x∈A and gx=x1/2 ∀x∈B Now, fgx=x122 fgx=x gfx=x212 gfx=x Since for gx=x1/2, x≥0 and range is also…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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