# composition of two functions

My friend has given me a math problem to solve just for fun. I have never seen these types of functions before but I undrestand what composition means.

If $f(x)=\left\{ \begin{array}{ll} 1+x, & \hbox{when }x<0 \\ 2 & \hbox{when }x>0 \end{array} \right.$ and $g(x)=\left\{ \begin{array}{ll} -1, & \hbox{when }x<1; \\ x, & \hbox{when }x\geq 1 \end{array} \right.$

How would I go about finding the formula of $g\circ f$ (composition of $g$ and $f$)?

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It's hard to tell that you're talking about piece wise functions since some of the line breaks are lost in formatting. I would edit, but I'm on iOS... :) – The Chaz 2.0 Mar 14 '12 at 6:47
What is $f(0)$? – anon Mar 14 '12 at 6:55

Determining $g\circ f\;$ reduces to characterizing when $f(x)\ge1$ or $f(x)<1$. Obviously $2\ge1$ so for any positive $x>0$ (where $f(x)=2$) we have $g(f(x))=f(x)=2$. Otherwise, when $x<0$, $f(x)=1+x$ is less than $1$ so $g(f(x))=-1$. You haven't prescribed what $f(0)$ is so I omit that for now.
$$(g\circ f)(x)=\begin{cases}2 & x>0 \\ -1 & x<0. \end{cases}$$