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Consider the function

$$ f(x_1, x_2) = m_1 x_1 + m_2 x_2 + b \,, $$

where $m_1, m_2 \in \mathbb{R}$ and $b \in \mathbb{R}$ are parameters.

Which of the following is correct:

  1. The function $f$ depends linearly on $x = (x_1, x_2)$.
  2. The function $f$ depends affine-linearly on $x = (x_1, x_2)$.

Usually I would call $f$ an affine-linear function.

But if I use this phrase, i.e. $f$ depends on $x$, I am not sure anymore.

I understand that this is a not too important question and probably for many both phrases are fine. Nevertheless, which of the two is better i.e. more precise?

Thank you in advance.

mathsstudentTUD
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  • Since $f(x) = (m+1)x$, I would say that $f$ is linear and, that $f$’s value depends linearly on $x$. –  Oct 21 '22 at 05:46
  • Thanks for your answer....Unfortunately I made a typo in the queation I asked. What I meant is $f(x) = mx+b$...I am sorry for this inconvience. – mathsstudentTUD Oct 21 '22 at 06:36
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    I would still say $f$ depends affine-linearly on $x$ (because we’re only dealing with $x$ as a variable by itself, so the goal is to emphasize the behavior of $f$ as a whole). But I see your point. For example, if we consider a function like $f:\Bbb{R}^2\to\Bbb{R}$, $f(x,t)=x^2+e^t$, then I would say something like $f$ depends quadratically on $x$ and exponentially in $t$ (I would say the same even if the function was $f(x,t)=x^2e^t$ etc). Therefore, to avoid such ambiguities, just go with “$f$ is an affine function” :) – peek-a-boo Oct 21 '22 at 06:49
  • Thank you very much for your detailed comment...I'm glad I'm not the only one who (has to) ponder over such things.... – mathsstudentTUD Oct 21 '22 at 06:58
  • Here is a related thread, what is the difference between linear and affine function?. – Apass.Jack Oct 21 '22 at 18:24

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