Question

What is the inverse function of ${{x}^{3}}$ ?

Answer 1

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.