Question

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

Answer 1

We will equate the above function with a variable as the inverse of that variable gives us the same function.
Then we will find the value of x from the equated equation. Later we will get the inverse function as $x$ will be equal to the inverse function in $y$.

Formula Used: If a function $f\left( x \right) = y$ , then it implies the inverse function also:
$x = {f^{ - 1}}\left( y \right)$

Complete step-by-step answer:
Let's say the above function is defined as $f\left( x \right) = y$.
Then the inverse of the function would be ${f^{ - 1}}\left( y \right) = x$.
But it is given that, $f\left( x \right) = 1 - {x^3}$.
So, it should be,
$\Rightarrow y = 1 - {x^3}$
Now, subtracting $1$ from both the side, we get the following equation:
$\Rightarrow y - 1 = 1 - {x^3} - 1$
Simplify the terms,
$\Rightarrow y - 1 = - {x^3}$
Now multiply both the sides by $- 1$, we get
$\Rightarrow 1 - y = {x^3}$
Now, taking the cube root on both sides, we get the following equation:
$\Rightarrow \sqrt[3]{{1 - y}} = x$
And, now if we tally with the above equation, we can derive the following equation:
$\Rightarrow x = {f^{ - 1}}\left( y \right) = \sqrt[3]{{1 - y}}$
Now, if we replace the value of $y$ by $x$ then we can say that,
$\therefore {f^{ - 1}}\left( x \right) = \sqrt[3]{{1 - x}}$

Hence, the inverse of $f\left( x \right) = 1 - {x^3}$ is $\sqrt[3]{{1 - x}}$.

Note:
The inverse of a function is a function that is reverse or reciprocal of that function.
If the function $f$ applied to an input $x$ gives a result of $y$, then applying its inverse function $g$ to $y$ will give us the result of $x$.
Always remember that the inverse of a function is denoted by ${f^{ - 1}}$.
Some properties of a function:
There is an always symmetry relationship exists between function and its inverse, that is why it states:
${\left( {{f^{ - 1}}} \right)^{ - 1}} = f$
If an inverse function exists for a given function then it must be unique by its property.