Let f be a function satisfying $$f(xy)=f(x)y$$ for all positive real numbers x and y . If $$f(30)=20$$ , then the value of $$f(40)$$ is

Functions (XII)

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$Let f be a function satisfying f (x y) = \frac{f (x)}{y} for all positive real numbers x and y . If$ $f (30) = 20, then the value of f (40) is$

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Solution

An equation of this kind is called a functional equation, and can often be solved by choosing particular values for the variables.
In this case, by choosing we see that
$\begin{array}{l} f (y) = \frac{f (1)}{y} for all y . Put y = 30; f (1) = 30 . f (30) = 30.20 = 600 . \\ Now f (40) = \frac{f (1)}{40} = \frac{600}{40} = 15 \end{array}$

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Functions (XII)

Remember concepts with our Masterclasses.

Let f be a function satisfying f(xy)=f(x)y for all positive real numbers x and y . If f(30)=20 , then the value of f(40) is

An equation of this kind is called a functional equation, and can often be solved by choosing particular values for the variables. In this case, by choosing we see that f(y)=f(1)y for all y. Put y=30;f(1)=30.f(30)=30.20=600. Now f(40)=f(1)40=60040=15

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$Let f be a function satisfying f (x y) = \frac{f (x)}{y} for all positive real numbers x and y . If$ $f (30) = 20, then the value of f (40) is$