Problem 1

Find all functions

f

defined on the set of positive real numbers which take positive real values and satisfy the two conditions: (i)

f(xf(y)) = yf(x)

for all positive real numbers

x, y

; (ii)

f(x) \to 0

x \to \infty

Step 6 of 6: Conclusion:

f(x)=1/x

Previous step Next step

f(xf(x))=xf(x)=1 \ \Longrightarrow\ f(x)=\dfrac1x

Detailed analysis

Setting $y=x$ in (i) gives $f(xf(x))=xf(x)$ , so $xf(x)$ is always a fixed point of $f$ . By Step 5 the only fixed point is $1$ , so $xf(x)=1$ for every $x>0$ , i.e. $f(x)=1/x$ . Conversely $f(x)=1/x$ satisfies both conditions: $f(xf(y))=f(x/y)=y/x=yf(x)$ , and $f(x)=1/x\to0$ as $x\to\infty$ . So $f(x)=1/x$ is the unique solution.