3.3 Linearity

A linear function, we have seen is a function whose graph lies on a straight line, and which can be described by giving the slope and y intercept of that line.

There is a special kind of linear function, which has a wonderful and important property that is often useful. These are linear functions whose \(y\) intercepts are \(0\) (for example functions like \(3x\) or \(5x\)). This means their graphs pass right through the origin, (the point with coordinates \((0, 0)\)). Such functions are called homogeneous linear functions. They have the property that their values at any combination of two arguments is the same combination of their values at those arguments. In symbols this statement is:

\[f(ax + bz) = af(x) + bf(z)\]

Do ordinary linear functions have any such property?

They sort of do. Any linear function at all has the same property when \(b\) is \(1 -a\). Thus for any linear function at all we have

\[f(ax + (1 - a)z) = a f(x) + (1 - a) f(z)\]

But be careful, linear functions that are not homogeneous do not obey the general linearity property stated several lines above.

Either one of these conditions allows you to calculate the value of \(f\) at any \(y\) given its value at \(x\) and \(z\). If \(y\) is \(z + a(x-z)\) then \(f(y)\) is \(f(x) + a(f(x)-f(z))\).

Properties like these mean that once you know the value of a linear function at two arguments you can easily find its value anywhere else it is defined.

The property here described is often called the property of linearity. This is not really a sensible way to describe it because perfectly good linear functions which have \(y\) intercept that is not \(0\) do not obey the more general version of the property (the first one above.)

Anyway, realize that functions that are not linear DO NOT have either of these properties.