Re: is f(x) = ax+b a linear function?
- From: kilian heckrodt <kilianheckrodt@xxxxxxxxx>
- Date: Sun, 08 Oct 2006 07:20:18 +0200
Kobu wrote:
Is f(x) = ax + b (where b, a <> 0) a linear function?
Everyone calls this linear but check this out:
f(y) = ay + b
f(z) = az + b
f(y+z) = a(y+z) + b = ay+ az + b <> f(y) + f(z) = ay+ az + 2b
f(cy) = a(cy) + b = cay + b <> cf(c) = cay + cb
f(x) fails both conditions for linearity. So why is it called linear
function?
You are absolutely correct, the term linear has a somewhat different meaning here.
f(x)=ax+b is linear as in describing a line (or linear growth), but it
is not linear in the sense of the formal definition of a linear function/map (i.e. satisfying f(x+y)=f(x)+f(y)).
Only for b=0 would f be a linear function in the latter sense.
Unfortunaly some mathematical terms can have rather different meanings
depending on the context, which is often due to parallel or historic
development of conventions.
.
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