Question

How do you write the equation for the inverse of the function $y=\arcsin \left( 3x \right)$?

Answer 1

Hint: In this problem we need to calculate the inverse function of the given function. For this we will calculate the value of $x$ from the given equation by applying the suitable reverse or inverse operations or function for the operations or functions we have in the given equation. We can observe that the function $\arcsin $ on the right side, so we will apply the reverse or inverses function $\sin $ on both sides of the given equation. Now we will simplify the obtained equation. After that we can observe that $3$ is in multiplication to $x$ on the right side. So, we will divide the obtained equation with $3$ on both sides of the equation and simplify the equation to get the value of $x$. Now we will replace the $x$ with ${{f}^{-1}}\left( x \right)$ and replace $y$ with $x$ to get the required result.

Complete step-by-step solution:
Given function is $y=\arcsin \left( 3x \right)$.
In the above equation we can observe $\arcsin $function on the right side. To get the $x$ value we are going to apply the inverse function to $\arcsin $ which is $\sin $ on both sides of the above equation, then we will get
$\Rightarrow \sin \left( y \right)=\sin \left( \arcsin \left( 3x \right) \right)$
We know that the value of $\sin \left( \arcsin x \right)=x$. From this formula the above equation is modified as
$\Rightarrow \sin y=3x$
In the above equation we can observe that $3$ is in multiplication. So, we are going to divide the above equation with $3$ on both sides, then we will get
$\begin{align}
  & \Rightarrow \dfrac{\sin y}{3}=\dfrac{3x}{3} \\
 & \Rightarrow x=\dfrac{\sin y}{3} \\
\end{align}$
Now replacing the $x$ with ${{f}^{-1}}\left( x \right)$ and $y$ with $x$, then we will get
$\Rightarrow {{f}^{-1}}\left( x \right)=\dfrac{\sin x}{3}$

Note: We can also check whether the obtained solution is correct or write by calculating the value of $f\left( {{f}^{-1}}\left( x \right) \right)$. We know that the value of $f\left( {{f}^{-1}}\left( x \right) \right)$ will be $x$ since the function $f$ and inverse function ${{f}^{-1}}$ will get cancelled. So, when we calculate the value of $f\left( {{f}^{-1}}\left( x \right) \right)$ it should be equal to $x$ otherwise our solution is incorrect.