Originally Posted by

**Greengoblin** You're right for a composition $\displaystyle g(f(x)) $ f's range is the domain of g....so for example, let:

$\displaystyle

f:[-3,3]\rightarrow\mathbb{R}, f(x)=x+1

$

$\displaystyle

g:\mathbb{R}\rightarrow\mathbb{R}, g(x)=x^2

$

Then the range of f is $\displaystyle [f(-3),f(3)] = [-2,4]$ then since g is composed with f, (also written $\displaystyle g\circ f(x)$) it's domain is the range of f, [-2,4]. Then the range of g is $\displaystyle [g(-2),g(4)] = [(-2)^2,4^2] = [4,16]$

So we have $\displaystyle f:[-3,3]\rightarrow[-2,4]$ and $\displaystyle g:[-2,4]\rightarrow[4,16]$ and for a compostion $\displaystyle f_1\circ f_2\circ...\circ f_n(x)$ you need only know the domain of $\displaystyle f_1$ to know all other domains/ranges.

However, the best appraoch to graphing would be to write out the composition as a single expression, e.g: $\displaystyle g\circ f(x) = (x+1)^2$ then choose a set of points from your domain e.g: $\displaystyle [-3,3]\in\mathbb{Z}$, calculate a table of values, plot the points and interpolate them into a smooth curve...just as you would with the graph of a normal function.

EDIT: Hmmm.. can someone please tell me the LaTeX for the composition symbol, the little circle? I thoubht it was: \circ

... I also tried \o and \O, but they didn't work either.