Question

Question #bfe8f

Answer 1

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`

Answer 2

`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.