Question

How do you find the inverse of $f(x) = 2 - 2{x^2}?$

Answer 1

We will equate the above function with a variable as 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 inverse function in y.

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

Complete step-by-step solution:
Let's say the above function is defined as $f(x) = y$.
Then the inverse of the function would be ${f^{ - 1}}(y) = x$.
But it is given that, $f(x) = 2 - 2{x^2}$
So, it should be $y = f(x) = 2 - 2{x^2}$
Now, subtracting $2$ from both the side, we get the following equation:
$\Rightarrow y - 2 = 2 - 2{x^2} - 2$
Or, $y - 2 = - 2{x^2}$
Now multiply both the sides by $- 1$, we get:
$\Rightarrow 2 - y = 2{x^2}$.
Now divide both the sides by $2$, we get:
$\Rightarrow \dfrac{{2 - y}}{2} = {x^2}$
Now, taking the square root on the both sides, we get following equation:
$\Rightarrow \sqrt {\dfrac{{2 - y}}{2}} = x$
And, now if we tally with the above equation, we can derive the following equation:
$\Rightarrow x = {f^{ - 1}}(y) = \sqrt {\dfrac{{2 - y}}{2}}$.
If we replace the value of $y$ by $x$ then we can say that:
$\Rightarrow {f^{ - 1}}(x) = \sqrt {\dfrac{{2 - x}}{2}}$.

$\therefore$The inverse of $f(x) = 2 - 2{x^2}$ is $\sqrt {\dfrac{{2 - x}}{2}}$.

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 exist 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.