Weird definition

Algebra Level pending

Let f f be a function of x x defined such that f 2 x = x ( 2 f − 1 ) . \dfrac{f^2}{x} = x^{(2f-1)}.

Find x x if f ( x ) = 1 2 f(x) = \dfrac{1}{2} .


The answer is 0.25.

Akeel Howell by Akeel Howell , 22, USA

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1 solution

Akeel Howell
Jan 24, 2018

We have that f 2 x = x ( 2 f − 1 ) ⟹ f 2 = ( x f ) 2 \dfrac{f^2}{x} = x^{(2f-1)} \implies f^2 = (x^f)^2

So f ( x ) = x f ( x ) ⟹ 1 2 = x 1 / 2 f(x) = x^{f(x)} \implies \dfrac{1}{2} = x^{1/2} .

Squaring both sides gives x = 0.25 x = \boxed{0.25}

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