f(x) =x+1 linearity

[mateh]

Sorry for the trivial question but... how can

f(x) = x + 1

be linear if

f(x + x') = x + x' + 1

is different than

f(x) + f(x') = x + 1 + x' + 1 = x + x' + 2

Thank you

 

[Deleted member 4993]

Sorry for the trivial question but... how can

f(x) = x + 1

be linear if

f(x + x') = x + x' + 1

is different than

f(x) + f(x') = x + 1 + x' + 1 = x + x' + 2

Thank you

What does x' indicate in your equations? How is it related to x?

 

[mateh]

Hello and thank you for your reply.

I'm just talking about the additivity condition of the linearity property.
If we take f(x) = x + 1, for x = 1 we get y = 2 and for x' = 2 we have y' = 3... but for x + x' = 1 + 2 we don't have y + y' = 5 but we get 4 instead, so it seems it doesn't satisfy the condition...

Am I missing something?

 

[JeffM]

Hello and thank you for your reply.

I'm just talking about the additivity condition of the linearity property.
If we take f(x) = x + 1, for x = 1 we get y = 2 and for x' = 2 we have y' = 3... but for x + x' = 1 + 2 we don't have y + y' = 5 but we get 4 instead, so it seems it doesn't satisfy the condition...

Am I missing something?

In the Wikipedia article that you linked to, it explains that there are two different usages of the word "linear." Look under linear polynomials where it explicitly says that linear polynomials do not satisfy the additivity condition except under special circumstances.

 

[mateh]

Thank you for the punctualization.

So f(x) = x + 1 is an affine function more precisely, a linear trasformation + translation...