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. $y=(a/x)$?

  • 2
    $\begingroup$ A linear function is one that satisfies $f(\lambda x) = \lambda f(x)$, for all appropriate scalars $\lambda$, and $f(x+y) = f(x)+f(y)$. This presupposes that $\lambda x$ and $x+y$ make sense, of course. People often use the term linear for an affine function (constant plus a linear function). I'm not sure what you meant by an 'inversely proportional function'. $\endgroup$ – copper.hat Oct 3 '12 at 16:07
  • $\begingroup$ Thanks, I fixed the inversely proportional function - wrote it wrong the first time. $\endgroup$ – Bob Oct 3 '12 at 16:08
  • $\begingroup$ $\frac{a}{2x} \neq 2 \frac{a}{x}$. So $f(2x) \neq 2 f(x)$. $\endgroup$ – copper.hat Oct 3 '12 at 16:09
  • $\begingroup$ For avoiding of any confusion, always think about what's the domain and range of your function. $\endgroup$ – Jack Oct 3 '12 at 16:09

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: $$\frac{d}{dx}(af(x)+bg(x))=af'(x)+bf'(x)$$

  • $\begingroup$ 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. $\endgroup$ – Stefan Smith Oct 3 '12 at 22:35

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.