# Find the domain of the following composite function. : f(x)= `sqrt(x-2)` g(x)= 1`/x` (g o f)(x)

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Given `f(x)=sqrt(x-2),g(x)=1/x` find the domain of `(g circ f)(x)` :

The domain of the composite function is the set of all x such that x is in the domain of f(x) and f(x) is in the domain of g(x).

The domain of f(x) is `x>=2` (assuming f(x) is a real-valued function) and the domain of g(x) is `x != 0` .

The domain of `g(f(x))` is all x in the domain of f(x) so `x>=2` such that f(x) is in the domain of g(x). So `sqrt(x-2)!=0 ==> x-2!=0 ==> x!=2`

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The domain of g(f(x))is x>2

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`g(f(x))=1/sqrt(x-2)` : the graph:

The function `f(x) = sqrt(x-2)` and `g(x) = 1/x`

`gof(x) = g(f(x)) = g(sqrt(x - 2)) = 1/(sqrt(x - 2))`

The domain of a function f(x) is the set of values of x for which f(x) is real and defined.

y = `1/sqrt(x - 2) `

The square root of a negative number is complex. Therefore `x - 2 >= 0` . Also, x cannot equal 2 as the value of 1/0 is not defined.

This gives the domain of gof(x) as `(2, oo)`