What is the inverse function of #F(x) = (7/x) - 3 #?

1 Answer

#F^{-1}(x)=7/(x+3)#

Explanation:

To invert an (invertible!) function #f# we can proceed as follows:

  1. Introduce a dependent variable #y# defined by the expression of the function in the independent variable #x#:
    #y:=f(x)#.

  2. Switch the roles of the dependent variable #x# and the independent variable #y#, trying to express #x# as a function of #y#. That function is called inverse function of #f# and it's denoted by #f^{-1}#:
    #x=f^{-1}(y)#

In our specific case, #F(x)=7/x-3#. We write
#y:=F(x)=7/x-3#
and now we solve for #x#:
#y+3=7/x#
#(y+3)/7=1/x#
#7/(y+3)=x#
We conclude that the inverse function of #F# is the function
#F^{-1}(y)=7/(y+3)#