Hi, I'm trying to understand the example the book gives:
For the functions and , state why is not defined.
Range of domain of g.
Therefore, g(f(x)) is not defined.
How can it not be defined? I've drawn graphs of and they look perfectly reasonable. What is going on? Thanks
Say is defined on domain . And is defined on domain . Suppose we can compose them in order for this expression to make sense: first we need that because we evaluate at first; then we want because we evaluate at second. So the rule is for to be defined we require that the range (output values) of to be contained in , the domain .
Thanks everyone,
So... the composition is 'undefined', even though there are only a few points where it is actually undefined out of infinity. Well I guess does contain infinity points... but that argument seems a bit strained.
So if , then g(f(x)) would be defined, right?
Ok how about this:
?
(My book says it's , but I don't believe this)
Sine , so i think you can only choose from the subset for the domain of the .
Then can be determined by only graphing the values ?
I think I'm finally getting my head around this, thanks so much for all your help.
no
Your book is right. A visualisation that might help you is an g-machine, into which you put in a number that is inside dom g, and an f-machine, into which you put a number which is inside dom f. The machine will then print out a number that is inside its range. f o g is the machine made when you attach the input of f to the output of g. It will only work if ran g dom f (you wrote this the wrong way round in your post). Otherwise g may print out a number that f can't handle.
Since ran g dom f, the new machine can handle any input in dom g.
what confused and irritated me when I was learning this is: Why don't we make a machine that works even if ran g dom f by making a smaller domain so that the machine only receives input that will produce a g(x) that is inside dom f?
The answer is that we do, but we don't call it defined.
I think I'm finally getting my head around this, thanks so much for all your help.