Composite Function Domains

[JamieLeaf]

Hi Guys,
I am really confused with the domain of composite functions. If we have a composite function f(g(x)), my textbooks says that the domain of this composite function is equivalent to the domain of g(x). However, after going online, I have found out that the domain needs to be worked out by comparing the domain of the composite function and the domain of the inside function. Could someone please tell me how you actually determine the domain of a composite function? Thanks!!

[DinWell]

Hi Guys,
I am really confused with the domain of composite functions. If we have a composite function f(g(x)), my textbooks says that the domain of this composite function is equivalent to the domain of g(x). However, after going online, I have found out that the domain needs to be worked out by comparing the domain of the composite function and the domain of the inside function. Could someone please tell me how you actually determine the domain of a composite function? Thanks!!

Your textbook is right in that that's always the case for a valid composite function,.
However, the method you use to find the domain will differ depending on the context of the question. If they give a composite function, f(g(x)) and they say that it exists, then you already know that the range of g(x) is a subset (or equal to) the domain of f(x). In this case, the domain will just be the domain of g(x). If they give two functions, f(x) and g(x) and tell you to find the domain such that f(g(x)) exists, then you have to find the domain of g(x) such that it's range is a (subset of / equal to) the domain of f(x). Then the domain of f(g(x)) will be the domain of g(x) that you found. You can remember this with the acronym RIDO (you've probably already learnt this), which stands for: Range Inner Domain Outer. Most of the time, you'd have to do the latter.

[dream chaser]

Hi Guys,
I am really confused with the domain of composite functions. If we have a composite function f(g(x)), my textbooks says that the domain of this composite function is equivalent to the domain of g(x). However, after going online, I have found out that the domain needs to be worked out by comparing the domain of the composite function and the domain of the inside function. Could someone please tell me how you actually determine the domain of a composite function? Thanks!!

Hi JamieLeaf,

Hopefully this will help. Lets consider the composite function to be fog. By the way, fog is the same as f[g(x)].

First you would draw a table(3x3) which has the range and domain of f(x) and g(x). By doing this, it makes it easier to understand as to what to do next.

Now, we have to check if fog exists. To see if it exists, the range of g(x) has to be equal or be a subset to the domain of f(x). If it does, this means that fog exists.

Now, that you know that fog exists, to see the domain of fog, you use one of the properties of composite functions where the domain of fog = domain of g(x).

Then, to find the range of fog, you draw that particular composite function with the domain of fog obtained in the previous step.

The same thing would apply if it was gof. However, to check to see if it exists, the range of f(x) has to be equal or be a subset to the domain of g(x). If it does, then the domain of gof = domain of f(x).

This is why drawing a table first is useful. It makes it easier to understand whether a composite function exists or not. However, if a composite function does not exist, and the composite function is fog, then you would have to alter the domain and range of f(x) until you make it exist. Same thing applies to gof and if it does not exist. You would then alter the domain and range of g(x) until you make it exist.

I hope this piece of advice helps.