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

Jul 13, 2017

Given: $y = \arcsin \left(3 x\right)$

Change to $f \left(x\right)$ notation:

$f \left(x\right) = \arcsin \left(3 x\right)$

Substitute ${f}^{-} 1 \left(x\right)$ for every x:

$f \left({f}^{-} 1 \left(x\right)\right) = \arcsin \left(3 {f}^{-} 1 \left(x\right)\right)$

The left side becomes x by definition:

$x = \arcsin \left(3 {f}^{-} 1 \left(x\right)\right)$

Use the sine function on both sides:

$\sin \left(x\right) = \sin \left(\arcsin \left(3 {f}^{-} 1 \left(x\right)\right)\right)$

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

$\sin \left(x\right) = 3 {f}^{-} 1 \left(x\right)$

Divide the equation by 3 and flip:

${f}^{-} 1 \left(x\right) = \sin \frac{x}{3}$

Before we can declare this as an inverse, we must test that $f \left({f}^{-} 1 \left(x\right)\right) = x$ and ${f}^{-} 1 \left(f \left(x\right)\right) = x$

$f \left({f}^{-} 1 \left(x\right)\right) = \arcsin \left(3 \left(\sin \frac{x}{3}\right)\right)$

$f \left({f}^{-} 1 \left(x\right)\right) = \arcsin \left(\sin \left(x\right)\right)$

$f \left({f}^{-} 1 \left(x\right)\right) = x$

${f}^{-} 1 \left(f \left(x\right)\right) = \sin \frac{\arcsin \left(3 x\right)}{3}$

${f}^{-} 1 \left(f \left(x\right)\right) = \frac{3 x}{3}$

${f}^{-} 1 \left(f \left(x\right)\right) = x$

Verified ${f}^{-} 1 \left(x\right) = \sin \frac{x}{3}$