Urgent: fog and gof

[Dark Horse]

Hey guys, I'm just a bit confused about finding the domain and range of fog and gof functions. I know how to prove that they exist, but dont know how to work out the domain and range of the final function.

eg. If you have fog, than is the range of fog the range of f, and the domain of fog the domain of g? Or do you have to sketch the graph of fog to work out its domain and range?

Thanks!

[ZachCharge]

The domain of f(g(x)) is the same as g(x).

[m@tty]

The domain of f(g(x)) is the same as g(x).

That's right, assuming that fog(x) is defined using that domain. Then use the range of g(x) as the domain of f(x) and calculate the range of f(x); this is the range of fog(x).

[Toothpaste]

Existence of the composite function
A composite function will only exist if the range of the "second" function is equal to or is part of the domain of the first.

Remember:
for or to be defined the range of g domain of f
for or to be defined the range of f domain of g


i.e. "inner's" range belongs to the "outer's" domain.


Domains of composite functions
If the composite function exists:
New domain of the composite function = Domain of the inside function

Remember:
The domain of  = domain of
The domain of  = domain of

If you're asked to find the range of the composite function, you would use the new domain to find it. Since there isn't a set rule for finding the range like the domain.

Using your question as an example

{0}

= =

Domain
would have the same domain as  since that is the "inner" function.
So the domain of is {0}

Range
The range of can be found using the domain:  {0}
= , it's linear as you can see. We want the range between domain: . At 0, = -4.

So therefore the range is .

Proving existence  
For to exist, the range of f must be a subset "" of the domain of g.

It's easier at first to draw up a table similar to this (but prettier):

Code: [Select]

    domain        range
f(x)    R         [-4,infinite)
g(x)   R+U{0}     [0,infinite)


• The range of is .
  The domain of is . (same as {0} )

So for to exist, must be a subset of .

BUT it's not, since the [-4, -1] extra doesn't belong in :

. So it's undefined, i.e. doesn't exist.

[the.watchman]

The domain of f(g(x)) is the same as g(x).

That's right, assuming that fog(x) is defined using that domain. Then use the range of g(x) as the domain of f(x) and calculate the range of f(x); this is the range of fog(x).

Or if you've found fog(x), then use Dom g to work out the range.
But I like your way m@tty

[Dark Horse]

so wait,

If you have fog, and it exists, then the domain of fog = domain of g, but to work out the range of fog, you cant say that its the range of f; you work it out by using the domain of fog to work out the range of fog. Coud someone please clarify this? And I dont get the Proving Existence bit: shouldnt that be worked out at the start? Thanks

[m@tty]

Yes, you must ensure that fog exists before you do anything with the domain or range. If it does not exist, ie. , then you must create a restriction in such that it is true. Generally (they want)/(you should find) the maximal domain, that is the largest domain in which it is defined.

fog(x) is saying take x and put it through g(x), then take this value and put it through f(x). It is like two machines, as shown in the picture below(note that is showing gof).

So once you know you fog is defined with a certain domain take that and put it through fog, or through g then the output of g through f; both will produce the same result.

[the.watchman]

YES! Dom fog = Dom g
And you would use Ran g in f(x) to work out Ran fog

And indeed, proof of existence should definitely go at the start, then you know whether the question can be answered or not

(Nice diagram m@tty )