Graphing Composite Functions

**zodiacbrave** · September 25th 2008, 10:03 AM

Hello, I am given two graphics, F(x), G(x) and I am supposed to graph g(f(x))

I am a bit rusty with composite functions, so I was hoping someone could give me some general information on how to do this? From what I recall, you need to take the range of f(x) and input it as the domain of g(x)? I know both functions domains must be satisfied. It looks like f(x) D = [-3,3] R = [-1,3] and g(x) D = [-3,3] R = [-2, 2]

I hope this made sense, thank you

**Greengoblin** · September 28th 2008, 08:48 AM

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

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

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

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

So we have $f:[-3,3]\rightarrow[-2,4]$ and $g:[-2,4]\rightarrow[4,16]$ and for a compostion $f_1\circ f_2\circ ...\circ f_n(x)$ you need only know the domain of $f_n$ 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: $g\circ f(x) = (x+1)^2$ then choose a set of points from your domain e.g: $[-3,3]\in\mathbb{Z}$ , 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.

**Moo** · September 28th 2008, 12:52 PM

Originally Posted by Greengoblin

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

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

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

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

So we have $f:[-3,3]\rightarrow[-2,4]$ and $g:[-2,4]\rightarrow[4,16]$ and for a compostion $f_1\circ f_2\circ...\circ f_n(x)$ you need only know the domain of $f_1$ 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: $g\circ f(x) = (x+1)^2$ then choose a set of points from your domain e.g: $[-3,3]\in\mathbb{Z}$ , 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.

Hi, yes it is \circ
but if you don't put any space between \circ and f, it will be understood as \circf, which is an unknown command

**Greengoblin** · September 29th 2008, 12:57 PM

Thanks, its working now. hoorah!

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