Is f(x) = ax + b Truly a Linear Function?

[pamparana]

Hello,

I was going through the same problem. From what I can see on the net, one of the relations a linear function has to express is:

f(u + v) = f(u) + f(v)

Now if f(x) = ax + b is linear then

f(u + v) = a(u+v) + 2b

So f(u + v) is not equal to f(u ) + f(v).

So why is f(x) = ax + b linear as it fails this criteria?

Sorry for this stupid question...

Thanks,
Luc

 

[wisvuze]

Hello,

I was going through the same problem. From what I can see on the net, one of the relations a linear function has to express is:

f(u + v) = f(u) + f(v)

Now if f(x) = ax + b is linear then

f(u + v) = a(u+v) + 2b

So f(u + v) is not equal to f(u ) + f(v).

So why is f(x) = ax + b linear as it fails this criteria?

Sorry for this stupid question...

Thanks,
Luc

It is not a stupid question. The definition above is exactly the definition of a linear function, i.e. a function f that satisfies f( cx + y ) = c f ( x ) + f ( y ) ( note that you can pull the scalar out, so that functions of the form cx are linear )
the map f ( x ) = ax + b is taught to us in grade school as a linear function on the basis that it draws a line. Actually though, it is called an "affine function" ( it acts essentially like a linear map though, the map will satisfy all the properties if you simply translate everything by b ).
In the context of differential equations, both cases are known as "linear equations"