Linearity

Definition of linearity

Function L has additivity and Homogeneity.

Let's find whether some functions have linearity.

In these example, some functions don't have linearity,

f(x) = ax, a is in real number, has linearity.

All linear functions don't have linearity.

f(x) = ax + b, b is not 0, is a linear function but it doesn't have linearity.

If functions are linearity, we can analyze and expect easily.

Let's see the graph below.

In this graph, If we want to find quantity,

 we don't have to estimate quantity for length of time of 287.

We just estimate quantity for length of time of 142 and 145.

Because we can use additivity in the graph above.

f( 142+145 ) = f( 142 ) + f( 145 )

Another method, We just estimate quantity for only length of time of 1.

Because we can also use homogeneity in the graph above.

f( 287 ) = 287 f( 1 )

If a graph doesn't have linearity, we can't expect quantity easily.

Look at this graph, 

we are unable to easily assume a function of the graph,

so, we can't easily expect quantity of length of time of 287

before that time.