AI-তৈরি নমুনা উত্তর
দেওয়া আছে,f ( x ) = x − 1 , ( x ≥ 1 ) f(x) = \sqrt{x-1}, \quad (x \ge 1) f ( x ) = x − 1 , ( x ≥ 1 ) g ( x ) = x 2 + 2 g(x) = x^2 + 2 g ( x ) = x 2 + 2
(i) ( g ∘ f − 1 ) ( x ) (g \circ f^{-1})(x) ( g ∘ f − 1 ) ( x ) y = f ( x ) = x − 1 y = f(x) = \sqrt{x-1} y = f ( x ) = x − 1 ⟹ y 2 = x − 1 \implies y^2 = x - 1 ⟹ y 2 = x − 1 ⟹ x = y 2 + 1 \implies x = y^2 + 1 ⟹ x = y 2 + 1 ∴ f − 1 ( x ) = x 2 + 1 \therefore f^{-1}(x) = x^2 + 1 ∴ f − 1 ( x ) = x 2 + 1 x ≥ 0 x \ge 0 x ≥ 0
অতএব,( g ∘ f − 1 ) ( x ) = g ( f − 1 ( x ) ) = g ( x 2 + 1 ) (g \circ f^{-1})(x) = g(f^{-1}(x)) = g(x^2 + 1) ( g ∘ f − 1 ) ( x ) = g ( f − 1 ( x )) = g ( x 2 + 1 ) = ( x 2 + 1 ) 2 + 2 = (x^2 + 1)^2 + 2 = ( x 2 + 1 ) 2 + 2 = x 4 + 2 x 2 + 1 + 2 = x^4 + 2x^2 + 1 + 2 = x 4 + 2 x 2 + 1 + 2 = x 4 + 2 x 2 + 3 = x^4 + 2x^2 + 3 = x 4 + 2 x 2 + 3
(ii) ( g ∘ f ) − 1 ( x ) (g \circ f)^{-1}(x) ( g ∘ f ) − 1 ( x ) ( g ∘ f ) ( x ) = g ( f ( x ) ) = g ( x − 1 ) (g \circ f)(x) = g(f(x)) = g(\sqrt{x-1}) ( g ∘ f ) ( x ) = g ( f ( x )) = g ( x − 1 ) = ( x − 1 ) 2 + 2 = x − 1 + 2 = x + 1 = (\sqrt{x-1})^2 + 2 = x - 1 + 2 = x + 1 = ( x − 1 ) 2 + 2 = x − 1 + 2 = x + 1
ধরি, y = ( g ∘ f ) ( x ) = x + 1 y = (g \circ f)(x) = x + 1 y = ( g ∘ f ) ( x ) = x + 1 ⟹ x = y − 1 \implies x = y - 1 ⟹ x = y − 1 ∴ ( g ∘ f ) − 1 ( x ) = x − 1 \therefore (g \circ f)^{-1}(x) = x - 1 ∴ ( g ∘ f ) − 1 ( x ) = x − 1
উত্তর: ( g ∘ f − 1 ) ( x ) = x 4 + 2 x 2 + 3 (g \circ f^{-1})(x) = x^4 + 2x^2 + 3 ( g ∘ f − 1 ) ( x ) = x 4 + 2 x 2 + 3 ( g ∘ f ) − 1 ( x ) = x − 1 (g \circ f)^{-1}(x) = x - 1 ( g ∘ f ) − 1 ( x ) = x − 1