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I'm curious what the definition of a linear function really is? I always hear that a linear function is a "straight line." That really isn't a definition- just a result.

Based on the real definition, how is an inversely proportional function not linear (i.e. $y=(a/x)$?

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    For avoiding of any confusion, always think about what's the domain and range of your function.2012-10-03

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A function $f$ is linear iff $f(ax)=af(x)$ and $f(x+y) =f(x) + f(y)$. This is sometimes written as one rule in the form: $f(ax + by) = af(x)+bf(y)$

A line has the property, so do planes. I don't know if you've studied calculus but even the derivative is linear: $\frac{d}{dx}(af(x)+bg(x))=af'(x)+bf'(x)$

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    In precalc a function is often defined as linear if it can be written in the form $f(x)=mx+b$. But the definition in the comments and the answers above are always used in more advanced math.2012-10-03