How do you sketch composite functions?

[horizon]

Is there some general method or shape for these?
Or do you just basically work from a table of values, plot the points and join them up?

Also, how do you sketch products of functions? Any good methods?

Thanks. 

[francis8nho]

I think I saw this somewhere in Mao's summary note. It's somewhere here - search along the MM cas. Good luck

[panicatthelunchbar]

Really good question! Made me think too :/

f(g(x)) is defined if rang is a subset of (cant do the symbol) domf.
g(f(x)) is defined if ranf is a subset of domg.

For example, if f(x)=x^2 and g(x)=2x+1;
and domf=R, ranf=R+ U {0}
and domg=R, rang=R

f(g(x)) is defined since rang is a subset of domf.           f(g(x))=f(2x+1)
                                                                                      = (2x+1)^2
g(f(x)) is defined since ranf is a subset of domg.           g(f(x))=g(x^2)
                                                                                      =2x^2+1

[panicatthelunchbar]

cont...

When sketching composite functions, check that you have the correct domain.
If f(g(x)) is defined, then domf(g(x))=domg
If g(f(x)) is defined, then domg(f(x))=domf

EXAMPLE: Let f(x)=x^2-2x
                   g(x)=|x|

Sketch f(g(x)) and g(f(x)) over the maximal domains.
domf=R     domg=R
ranf=[-1,infinity)     rang=R+ U {0}

f(g(x)) is defined since rang is a subset of domf.
g(f(x)) is defined since ranf is a subset of domg
f(g(x))=f|x|
         =|x|^2 - 2|x|

The easiest way to sketch this graph is to sketch y=x^2 - 2x for x(greater than or equal to 0), and then keep this part of the graph as well as the reflection of the graph in the y-axis. ( I hope you have done modulus functions and their graphs)!

Answer in both parts.       g(f(x))=g(x^2-2x)
                                             = |x^2 - 2x|

The easiest way to sketch this is to sketch y= x^2 - 2x and reflect any sections below the x-axis along the x-axis, so that the whole graph is above the x-axis.

I really hope this was what you were after!

[cltf]

When you mean sketch composite functions do you mean the restricts that apply or for example f(x)=1/x and g(x)=Sqrt(x+3)therefore sketch f(g(x))?

[TrueTears]

Is there some general method or shape for these?
Or do you just basically work from a table of values, plot the points and join them up?

Also, how do you sketch products of functions? Any good methods?

Thanks. 

1. derive find stationary points
2. find x/y axis intercepts
3. find concavitivity of all critical points
4. done