F-0- - - 0- - fx fy - x2 ya a - R thenValue of a is

Put y =1 f(x f(1)) =x^2 Let f(1) =k, f(kx) =x^2 f(x) =x^2/k^2 f(1) = 1^2/k^2 = k ⇒ k^3 =1 k =1 So f(x)=x^2 f(x.y^2) =x^2 y^a ⇒ (xy^2)^2 =x^2 y^a x^2 y^4 =x^2 y^ ...

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Byju's Answer

Standard XI

Mathematics

Composite Function

f:0, ∞ → 0, ∞...

Question

$f : (0, \infty) \to (0, \infty), f (x f (y)) = x^{2} y^{a} (a \in R)$ then

Value of a is

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Solution

The correct option is B 4
Put y =1
$f (x f (1)) = x^{2}$
Let $f (1) = k, f (k x) = x^{2}$
$f (x) = \frac{x^{2}}{k^{2}}$
$f (1) = \frac{1^{2}}{k^{2}} = k \Rightarrow k^{3} = 1$
k =1
So $f (x) = x^{2}$
$f (x . y^{2}) = x^{2} y^{a} \Rightarrow (x y^{2})^{2} = x^{2} y^{a}$
$x^{2} y^{4} = x^{2} y^{a}$
$\Rightarrow a = 4$
For $2 f (x) = e^{x}$
$2 x^{2} = e^{x}$

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