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For this question the definition for monotone increasing is that $f$ is monotone increasing if for all $x_1<x_2$ where $x_1,x_2\in I$, $f(x_1) \leqslant f(x_2)$.

I have to apply that definition to $f$, $g$ and $f\circ g$ but I'm not quite sure exactly how.

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  • $\begingroup$ Why would the composite $f\circ g$ ever not be defined? $\endgroup$ – Mario Carneiro Jun 9 '15 at 3:38
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Hint:

$$x\le y\to g(x)\le g(y)\to f(g(x))\le f(g(y))$$

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  • $\begingroup$ i have that written down but im not quite sure how to tie it together i have defined each function f, g , and f o g using the definition and thats what i ended up with but is there a way that ties them together? $\endgroup$ – ChrisV Jun 9 '15 at 3:47
  • $\begingroup$ @Chris If you include $f\circ g(x)=f(g(x))$ and the definition of increasing (which you can easily show is equivalent to $x\le y\to f(x)\le f(y)$), you should have all the pieces you need. $\endgroup$ – Mario Carneiro Jun 9 '15 at 3:57

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