Question

How do you write the equation for the inverse of the function `y=arcsin(3x)`?

Answer 1

Given: `y=arcsin(3x)`

Change to `f(x)` notation:

`f(x)=arcsin(3x)`

Substitute `f^-1(x)` for every x:

`f(f^-1(x))=arcsin(3f^-1(x))`

The left side becomes x by definition:

`x=arcsin(3f^-1(x))`

Use the sine function on both sides:

`sin(x)=sin(arcsin(3f^-1(x)))`

Because the sine and the arcsine are inverses, they cancel:

`sin(x)=3f^-1(x)`

Divide the equation by 3 and flip:

`f^-1(x)= sin(x)/3`

Before we can declare this as an inverse, we must test that `f(f^-1(x)) = x` and `f^-1(f(x)) = x`

`f(f^-1(x)) = arcsin(3(sin(x)/3))`

`f(f^-1(x)) = arcsin(sin(x))`

`f(f^-1(x)) = x`

`f^-1(f(x)) = sin(arcsin(3x))/3`

`f^-1(f(x)) = (3x)/3`

`f^-1(f(x)) = x`

Verified `f^-1(x)= sin(x)/3`