Composte functions help.

[tolga]

I can find the domain and range for each but have trouble identifying whether it is defined when comparing the domain and range and what Q2 is tryning to say.
Q1) f(x)=root(x-2) , g(x)=1/x+1  +2 , determine whether f(g(x)) exists and g(f(x)) exists and if so, find the composition functions.

Q2 f(x)=3-root(x) and g(x)= x^2  -1, show that f(g(x)) is not defined. by restricting the domain of g, find a function h such that f(h(x)) is defined.

Q3  f(x)=1/(x+a)^2 and g(x)=root(x), determine values of a such that f(g(x)) exists.

[the.watchman]

The domain of a composite function is the domain of the 'inside' function (in these cases, g)
The range can be determined from the domain.

Basically, for Q2, because , you need to restrict the range of to allow the composite function to be defined:

In this case, Dom f = , so Ran g must be within this, so the new Dom g can become OR

[tolga]

sorry but i dont understand anything you are sayin

[the.watchman]

sorry but i dont understand anything you are sayin

Erm...sorry! I'll try again...

To find the domain of a composite function, you work out the domain of the function inside f(x), in this case it is g(x). The domain of the composite is equal to this.

To do Q2, you need to make f(g(x)) defined, so you will need to restrict the domain of g(x)
Now considering that, for a composite function (such as f[g(x)] ) to be defined, the range of g must be within the domain of f.
In this case, it is not, because [-1,infinity) is not within [0,infinity)

Therefore, you need to find the instance when the range of g is within [0,infinity)
If you visualise the graph of g, you will find that for the range of g to be within [0,infinity), the domain of g must be either (-infinity,-1] or [1,infinity)
Using this restricted domain, you can find f(h(x)) and it will be defined.

Hope this is better