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I have the following problem:

$f: \mathbb R\to +\infty $ and $f(x) - f(y) = f( x / y) , x,y > 0 $

a)Show that $f(1) = 0$ b)Show that $f$ is one-to-one and that $f(x)=0$ has a single solution

c)solve the equation $ f(x^2 -2) + f(x) = f(5x -6) $

d) If $f(x) > 0$ for every $x>1$, show that $f$ is strictly increasing at $(0, +\infty) $

Thank you

  • 1
    $f:\mathbb{R}\to +\infty$ what does it mean?2012-11-18

1 Answers 1

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$(a)\,\,\;\;\;\;f(1)=f\left(\frac{1}{1}\right)=f(1)-f(1)$

$(b)\;\;\;\;\;x\neq y\Longrightarrow f(1)=0\neq f(x)-f(y)=f\left(\frac{x}{y}\right)\Longrightarrow \frac{x}{y}\neq 1$

Take it from here...