Specialist's Methods Thread

[kamil9876]

f(g(x)) can't exist (since 10.1 is in the range of g but f can't 'eat' 10.1 (it is outside its domain))

for similair reasons g(f(x)) can't exist.

[pi]

f(g(x)) can't exist (since 10.1 is in the range of g but f can't 'eat' 10.1 (it is outside its domain))

for similair reasons g(f(x)) can't exist.

To format in a VCAA exam, draw a table for ease

            |     f(x)     |     g(x)     
Ran      |     ...       |      ...         
Dom     |     ...       |      ...         

Then say something like "as Domf isn't a subset or equal to (use the symbol) Rang, g(f(x)) cannot exist"




edit: just looked like I corrected kamil, which I wasn't trying to do! Sorry if it came out that way haha! kamil's right, it's just that assessor's like the above format

[Special At Specialist]

Okay, but I still don't understand. Can you please give me an example of where f(g(x)) does exist and show me how to find the domain and range of it?

[brightsky]

consider the diagram below for composite function f(g(x)) or fog:

x---> g-machine ---> g(x) ---> f-machine ---> f(g(x))

you grab a set of x-values, chuck them in the g-machine, and get a set of g(x) values. for f(g(x)) to exist, all g(x) values must be within the domain of f(x).

for example, if your f(x) was sqrt(x), and you input g(x) = -1 into it, you won't get any real output f(g(x)).

more formally, we say for f(g(x)) to exist, ran g must be a subset (or equal to) dom f. if f(g(x)) exists for all g(x), then it follows that it's domain is the set of all x, which is defined in this case as the domain of g.