Question #bfe8f

2 Answers
May 7, 2017

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#

May 7, 2017

Answer:

#f^-1(x)=1/sin^-1(7x-15)#

Explanation:

When #f(x)# is equil to the equation and you add it to the power of -1 then you actually just divide 1 by #f(x)#, thus meaning you should devide 1 with the equation as well. Another answer that would also be correct is just to take the entire equation to the power of -1
#f^-1(x)=(sin^-1(7x-15))^-1# wich should give you the same answer.