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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