View Full Version : (g of f)(x)
Alem2000
Jun7-04, 02:27 PM
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)
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.
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.
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.
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 :wink:
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