How do you find domain and range for f(x)= x/(x-2)?

1 Answer
Alan P.
Oct 1, 2015

Domain: RR - {2}
Range: RR - {1}

Explanation:

f(x)=2/x-2 is clearly defined for all Real values of x except x=2 (since division by 0 is undefined).
Therefore the Domain is all Real values except 2

To determine the range, consider the possible limitations on g(x) where g(x) is the inverse of f(x)

By definition of inverse
color(white)("XXX")f(g(x)) = x
and from the given definition of f(x)
color(white)("XXX")f(g(x)) = g(x)/(g(x)-2)

Therefore
color(white)("XXX")g(x)/(g(x)-2) = x

color(white)("XXX")g(x) = x*g(x) -x*2

color(white)("XXX")g(x)-x*g(x) = -2x

color(white)("XXX")g(x)(1-x) = -2x

color(white)("XXX")g(x) = (-2x)/(1-x)

which is defined for all values of x!=1

Therefore the Range is all Real values except 1

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