How do you find the inverse of #f(x) = x/(x+1)# and is it a function?

1 Answer
Alan P.
Apr 9, 2016

If #bar(f)(x)# is the inverse of #f(x)=x/(x+1)#
then
#color(white)("XXX")bar(f)(x)=x/(1-x)# (which is a function.)

Explanation:

If #color(red)(bar(f)(x))# is the inverse of #color(blue)(f(x))#
then
#color(white)("XXX")color(blue)(f"(")color(red)(bar(f)(x))color(blue)(")") = x#
and
#color(white)("XXX")color(blue)(f"(")color(red)(bar(f)(x))color(blue)(")")=(color(red)(bar(f)(x)))/(color(red)(bar(f)(x))+1)#

#color(white)("XXX")color(red)(bar(f)(x))=x*(color(red)(bar(f)(x))+1)#

#color(white)("XXX")color(red)(bar(f)(x))-x*color(red)(bar(f)(x)) =x#

#color(white)("XXX")color(red)(bar(f)(x))*(1-x)=x#

#color(white)("XXX")color(red)(bar(f)(x)) = x/(1-x)#

Since this expression provides a unique solution for all valid values of #x#; it is a function.

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