Given: #y=(3x-7)/(x+9)#
Write as a function of x:
#f(x)=(3x-7)/(x+9)#
Substitute #f^-1(x)# for every x:
#f(f^-1(x))=(3f^-1(x)-7)/(f^-1(x)+9)#
The property #f(f^-1(x)) = x# tells us that the left side becomes x:
#x=(3f^-1(x)-7)/(f^-1(x)+9)#
Multiply both sides by #f^-1(x)+9#:
#x(f^-1(x)+9)=3f^-1(x)-7#
Use the distributive property on the left:
#xf^-1(x)+9x=3f^-1(x)-7#
Add -3f^-1(x) - 9x to both sides:
#xf^-1(x)-3f^-1(x)= -9x-7#
Multiply both sides by -1:
#3f^-1(x)-xf^-1(x)= 9x+7#
Factor out #f^-1(x)#:
#(3-x)f^-1(x)= 9x+7#
Divide both sides by #3-x#
#f^-1(x)= (9x+7)/(3-x)#
One cannot declare that the above is #f^-1(x)# until one has checked that #f(f^-1(x)) = x# and #f^-1(f(x))=x#:
Start with:
#f(x)=(3x-7)/(x+9)#
Substitute #x = f^-1(x) = (9x+7)/(3-x)#:
#f(f^-1(x))=(3(9x+7)/(3-x)-7)/((9x+7)/(3-x)+9)#
#f(f^-1(x))=(3(9x+7)-7(3-x))/((9x+7)+9(3-x))#
#f(f^-1(x))=(27x+21-21+7x)/(9x+7+27-x)#
#f(f^-1(x))=(34x)/34#
#f(f^-1(x))=x larr# the first part checks.
Start with:
#f^-1(x)= (9x+7)/(3-x)#
Substitute #x = f(x) = (3x-7)/(x+9)#:
#f^-1(f(x))= (9(3x-7)/(x+9)+7)/(3-(3x-7)/(x+9))#
#f^-1(f(x))= (9(3x-7)+7(x+9))/(3(x+9)-(3x-7))#
#f^-1(f(x))= (27x-63+7x+63)/(3x+27-3x+7)#
#f^-1(f(x))= (34x)/34#
#f^-1(f(x))= x#
Both check, therefore, #f^-1(x)= (9x+7)/(3-x)#
