Many physical quantities are connected by inverse square laws, that is, by power functions of the form f(x)=kx^(-2). In particular, the illumination of an object by a light source is inversely proportional to the square of the distance from the source. Suppose that after dark you are in a room with just one lamp and you are trying to read a book. The light is too dim and so you move halfway to the lamp. How much brighter is the light?

Since the equation for the illumination of an object, i.e. the brightness of the light, is inversely proportional to the square of the distance from the light source, the form of the function is:

f(x) = k.x⁻²

Where x is the distance between the object and the light force, k is the constant of proportionality, and f(x) is the brightness.

Then, if you move halfway to the lamp the new distance is x/2 and the new brightness (call if F) is :

[tex] F=k(x/2)^{-2}=\frac{k}{(x/2)^2}= \frac{k}{x^2}. 4=f(x).4[/tex]

Then, you have found that the light is 4 times as bright as it originally was.