How do you find the inverse of #f(x)= 1/3x + 2# and is it a function?

1 Answer
Alan P.
Jul 18, 2016

#color(blue)(g(x)=3x-6)# is a function
(which is the inverse of #f(x)=1/3x+2#)

Explanation:

If #g(x)# is the inverse of #f(x)#
then by definition
#color(white)("XXX")g(f(x))=x#
and
#color(white)("XXX")f(g(x))=x#

Since #f(color(red)(x))=1/3color(red)(x)+2#
then
#color(white)("XXX")f(color(red)(g(x)))=1/3color(red)(g(x))+2#

But from our original definition we know that
#color(white)("XXX")f(g(x))=x#

Therefore
#color(white)("XXX")f(g(x))=1/3g(x)+2=x#

#color(white)("XXX")rArr1/3g(x)=x-2#

#color(white)("XXX")rArrg(x)=3x-6#

Since any value of #x# gives a single value for #g(x)#
it follows that #g(x)# is a function.

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