Given: #f(x) = sin^-1( 7x -15)#
Substitute #f^-1(x)# for every #x#:
#f(f^-1(x)) = sin^-1( 7f^-1(x) -15)#
The left side becomes #x# by definition:
#x = sin^-1( 7f^-1(x) -15)#
Take the sine of both sides:
#sin(x) = 7f^-1(x) -15#
Add 15 to both sides:
#sin(x)+15 = 7f^-1(x)#
Divide both sides by 7:
#f^-1(x)= (sin(x)+15)/7#
Before one can declare this as the inverse, one must show that #f(f^-1(x)) = x# and #f^-1(f(x)) = x#:
#f(f^-1(x)) = sin^-1( 7((sin(x)+15)/7) -15)#
#f(f^-1(x)) = sin^-1( sin(x)+15 -15)#
#f(f^-1(x)) = sin^-1( sin(x))#
#f(f^-1(x)) = x#
#f^-1(f(x)) = (sin(sin^-1( 7x -15))+15)/7#
#f^-1(f(x)) = ( 7x -15+15)/7#
#f^-1(f(x)) = (7x)/7#
#f^-1(f(x)) = x#
Q.E.D.
#f^-1(x)= (sin(x)+15)/7#
