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

Question

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 (A) 15 (B) 20 (C) 40 (D) 60

Tardigrade Admin · Accepted Answer

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

x = 1

, we see that

f (y) = \frac{f ( 1 )}{y}

for all y. put

y

= 30; f (1) = 30 \cdot f (30) = 30 \cdot 20 = 600

. Now f (40) = \frac{f ( 1 )}{40} = \frac{600}{40} = 15

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

15

20

40

60

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 x=1, we see that f(y)=yf(1)​ for all y. put y =30;f(1)=30⋅f(30)=30⋅20=600 . Now f(40)=40f(1)​=40600​=15

Q. 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