# If $f$ and $g$ are monotone increasing and if the composite function $f\circ g$ is defined, then it is also monotone increasing.

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.

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

$$x\le y\to g(x)\le g(y)\to f(g(x))\le f(g(y))$$
• @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. – Mario Carneiro Jun 9 '15 at 3:57