Question

How do you find the inverse of $f\left( x \right)=1-{{x}^{3}}$?

Answer 1

In this question we have the function as $f\left( x \right)$ for which we have to find the inverse function which is denoted as $f'\left( x \right)$. Inverse of the function is the image of the function reflected over the line $y=x$. In this question, we will change the function definition by considering $f\left( x \right)=y$ and then solving for the value of $x$, which will give us the required inverse function.

Complete step by step solution:
We have the given function as:
$\Rightarrow f\left( x \right)=1-{{x}^{3}}$
In this question, we will consider $f\left( x \right)$ and change the function definition. On substituting it in the equation, we get:
$\Rightarrow y=1-{{x}^{3}}$
On transferring the term $-{{x}^{3}}$ from the right-hand side to the left-hand side, we get:
$\Rightarrow y+{{x}^{3}}=1$
On transferring the term $y$ from the left-hand side to the right-hand side, we get:
$\Rightarrow {{x}^{3}}=1-y$
Now we want the function in terms of $x$ therefore, on taking the cube root on both the sides of the expression, we get:
$\Rightarrow \sqrt[3]{{{x}^{3}}}=\sqrt[3]{1-y}$
Now since the cube root of a term which is already in the cube format cancel out, we get the expression as:
$\Rightarrow x=\sqrt[3]{1-y}$
Now we will substitute the value of $y$ as $x$ so that we get the function in terms of $x$.
$\Rightarrow x=\sqrt[3]{1-x}$
Which is the required inverse function.
Therefore, we can write it as:
$\Rightarrow f'\left( x \right)=\sqrt[3]{1-x}$, which is the required solution.

Note: Inverse function is a function which reverses the value of the function. It is also called the anti-function. The basic steps to solve the inverse of a function should be remembered which is that every instance of $x$ should be replaced by $y$ and every instance of $y$ should be replaced with $x$, then solve for $y$ to get the inverse function.