# How do you find the inverse of f(x) = 5 sin^{-1}( frac{2}{x-3} )?

Mar 13, 2018

Well... you kind of can and cannot....

#### Explanation:

I will explain the method used for finding the inverse function but keep in mind that the answer is not an actual inverse function. The reason there is no inverse function is because $f \left(x\right) = \sin \left(x\right)$ is a many-to-one function meaning that several different values of x will give the same value of $f \left(x\right)$ and it is therefore impossible to trace a value back to a single value of x. However, you can have an inverse function if you restrict x to a certain domain. Note that this domain will be equal to the range of the inverse function.

When finding the inverse of a function it helps to think of functions in terms of x and y.

$f : y = 5 {\sin}^{- 1} \left(\frac{2}{x - 3}\right)$

where f is a function stating that y is equal to the right hand side.

Then you switch the x's and y's and write it as the inverse function for x.
${f}^{- 1} : x = 5 {\sin}^{- 1} \left(\frac{2}{y - 3}\right)$

However, to get the inverse function for y, we must rearrange the equation to get y on the left hand side along with ${f}^{- 1} :$ so that we get the expression ${f}^{- 1} : y =$ some value.

${f}^{- 1} : x = 5 {\sin}^{- 1} \left(\frac{2}{y - 3}\right)$
${f}^{- 1} : \frac{x}{5} = {\sin}^{- 1} \left(\frac{2}{y - 3}\right)$ // divide both sides by 5
${f}^{- 1} : \sin \left(\frac{x}{5}\right) = \frac{2}{y - 3}$ // $\sin$ is the inverse function of ${\sin}^{- 1}$

${f}^{- 1} : \frac{1}{\sin \left(\frac{x}{5}\right)} = \frac{y - 3}{2}$ // take the reciprocal of both sides

${f}^{- 1} : \frac{2}{\sin \left(\frac{x}{5}\right)} = y - 3$ // multiply both sides by 2

${f}^{- 1} : \frac{2}{\sin \left(\frac{x}{5}\right)} + 3 = y$ // add 3 to both sides

Therefore, we find the inverse function to be:

${f}^{- 1} : y = \frac{2}{\sin \left(\frac{x}{5}\right)} + 3$

But remember that for this to be an inverse function there must be a restricted domain making the original function a one-to-one function. A one-to-one function is a function where for each x there is only one value of y and vice versa