# (g of f)(x)

Questoin...I have allways done these by evaluating (F 0f G)(x) ...then finding the domain of G and then useing a graphing calculator to find the domain of the whole thing as a function. My question is how do you algebracially fine the domain of the function after you find the domain of G int the case of (F of G)(X). If you can show me using an example where f=square root of x-2 and g= 2-square root(x)

## Answers and Replies

Alem2000 said:
Questoin...I have allways done these by evaluating (F 0f G)(x) ...then finding the domain of G and then useing a graphing calculator to find the domain of the whole thing as a function. My question is how do you algebracially fine the domain of the function after you find the domain of G int the case of (F of G)(X). If you can show me using an example where f=square root of x-2 and g= 2-square root(x)
You can't find the domain of a function. It is usually given, implicitly or otherwise. Let h(x) = f(g(x)). What is the domain of h? What is the range? It should be obvious from the expression.

arildno
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These types of exercises should usually be interpreted as "what is the maximal domain this function can have?"
In the given exercise, we have:

$$f(x)=\sqrt{x-2},g(x)=2-\sqrt{x}$$

Hence, for f(g(x)), we need g(x)>=2

This means: $$2-\sqrt{x}\geq{2}\to\sqrt{x}<=0$$

Hence, only a single x-value is eligible as the domain of f(g(x)), namely x=0,
so that is the (maximal) domain.

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arildno said:
$$f(x)=\sqrt{x-2},g(x)=2-\sqrt{x}$$

Hence, for f(g(x)), we need g(x)>=2
...
But this depends on whether f is defined on the reals. If the range of f is a subset of the complex numbers, then the 'maximal domain' is just the domain. This brings me back to my previous post.

arildno
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Sure enough, but these exercises rarely show up again once students have reached the level of learning about complex numbers..
So, I stand by my answer as most probably correct 