Question

What is the inverse function of \[{{x}^{3}}\] ?

Answer 1

Hint: Inverse function is a function that reverses another function. Suppose if the function \[f\] is applied to an input \[x\] gives a result of \[y\], then applying the inverse function \[g\] to \[y\] gives the result \[x\] i.e. \[g\left( y \right)=x\] if and if only if \[f\left( x \right)=y\]. Inverse functions can be solved by replacing the variables. For example, if we have x and y in the function, then we can replace all \[x\] with \[y\] and all \[y\] with \[x\].

Complete step by step solution:
Now, let us find out the inverse function of \[{{x}^{3}}\].
Let us consider the given function to be \[y={{x}^{3}}\].
In order to find the inverse function of \[{{x}^{3}}\], we will take cube root on both sides as below,
\[{{y}^{\dfrac{1}{3}}}={{x}^{3\left( \dfrac{1}{3} \right)}}\]
To solve the above equation, we will cancel out 3 in the Right Hand Side. After solving, we will get,
\[{{y}^{\dfrac{1}{3}}}={{x}^{{}}}\]
Now in order to find out the inverse function, let us inverse both the terms on both the sides of the equation. On inversing the functions, we get
\[{{y}^{-1}}={{x}^{\dfrac{1}{3}}}\]
\[\therefore \] We have got the inverse function of \[{{x}^{3}}\] as \[{{x}^{\dfrac{1}{3}}}\].

Note: Let us check for some facts on inverse of a function. To find inverse of a function, it must satisfy a condition i.e. for a function \[f:X\to Y\] to have a inverse, it must have a property that for every \[Y\] in \[y\], there is exactly one \[x\] in \[X\] such that \[f\left( x \right)=y\]. This property ensures that a function \[g:Y\to X\] exists with the necessary relationship with \[f\].Consider a function \[f\], if the graph of \[f\] intersects at a horizontal line more than once, then we understand that there exists no inverse function to that function.