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If fog and gof are both linear functions, then must f or/and g also be linear functions? Proof or counterexample(s)?

If fog is linear, f and g does not have to be linear. Take f=g^-1. sin(arcsinx) or In(e^x).

mathwriter
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  • This isn't true. If $f(x) = 0$ forall $x$, and $g(0) = 0$, then $(f \circ g)(x) = 0$ and $(g \circ f)(x) = 0$ for all $x$, so $f\circ g$ and $g \circ f$ are both linear but unless you are working in the trivial vector space ${0}$ there are many non-linear $g$ such that $g(0) = 0$. (You haven't said what field of coefficients you are working with so I am guessing it is $\Bbb{R}$. If you are working over arbitrary fields then my last statement isn't quite right over $\Bbb{Z}/2$ - for which you need to assume that neither space is trivial and that one of them has at least $3$ elements.) – Rob Arthan Oct 31 '23 at 21:27
  • Thank you. Let us assume that we are working with the reals. Let (fog)(x)=2x+2 and (gof)(x)=x+1. Then, what can we say about the linearity of f or/and g? Are they also in the form of ax+b, where a and b are real numbers. – mathwriter Nov 01 '23 at 03:58
  • I think you are a bit confused about the definition of a linear function. Linear functions must map $0$ to $0$ so neither $2x+2$ nor $x+1$ is a linear function. Functions like $2x + 2$ add a constant ($2$) to a linear function ($2x$) are called affine. – Rob Arthan Nov 01 '23 at 21:37
  • Thanks. Then my question in general is as follows, considering real numbers. If fog and gof are affine, then are f or/and g also affine? – mathwriter Nov 01 '23 at 23:48
  • Answer: no see my first comment. – Rob Arthan Nov 02 '23 at 00:33

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