Question

If $a$ and $b$ are positive integers, $f$ is a function defined for positive numbers and attains only positive values such that $f\left(yf\right(x\left)\right)=x^a{y}^b$, then

• A. $b^3=a$
• B. $a^2=b$
• C. $b=a$
• D. $b^2=a$

Answer 1

The correct option is D $b^2=a$

$f\left(yf\right(x\left)\right)=x^a{y}^b...\left(1\right)$
Put $y=\frac{1}{f\left(x\right)}$
$f\left(1\right)=x^a\cdot \frac{1}{\left[f\right(x){]}^b}$
Put $x=1,$ we get $f\left(1\right)=1$
$\Rightarrow \left[f\right(x){]}^b=x^a$
$\Rightarrow f\left(x\right)=x^{a/b}\cdots \left(2\right)$

Now, Put $x=1$ in eqn $\left(1\right),$ we get
$f\left(y\right)=y^b(\because f(1)=1)$
Replcaing $y$ by $x$, we get
$f\left(x\right)=x^b\cdots \left(3\right)$

Using $\left(2\right)$ and $\left(3\right)$,
$x^b=x^{a/b}\Rightarrow b=\frac{a}{b}\Rightarrow b^2=a$