find the domain and range

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Ihope the solution can help you^^”

Simple ideas of domain and range

Domain of a real function y=f(x) is the set of real values of x for which y is well-defined,
i.e. y can be calculated.
Range of a real function y=f(x) is the set of values of y corresponding to the values of x in the domain.

A method to find domain and range of y=f(x):
Make y the subject, i.e. y=f(x), the set of values of x for which y is well-defined
is the domain.
Make x the subject if possible, i.e. x=h(y), the set of values of y for which x is
well-defined is the range

Solution:

Let y=g(x). Then y =1/ [1-√(x-2)].

As denominator cannot be zero and square root of negative number is unreal,
y is well-defined if 1-√(x-2)≠0 and (x-2)≥0
∴x≠3 and x≥2
2≤x<3 or x>3
Domain is 2≤x<3 or x>3.

Make x the subject:
y=1/ [1-√(x-2)]
y[1-√(x-2)]=1
y-y√(x-2)=1
y-1=y√(x-2)
x-2=(y-1)²/y²
x=(y-1)²/y²+2
x is well-defined if y≠0.
Range is x≠0. (Either y≠0 or x≠0 means all non-zero numbers)