Question

How do you find ${{f}^{-1}}\left( x \right)$ given $f\left( x \right)=\dfrac{1}{{{x}^{3}}}$

Answer 1

This type of problem is based on the concept of finding inverse for a function. First, we have to assume the given function as y, that is, $f\left( x \right)=y$. Then, make necessary calculations and find the value of x which will be in terms of y by taking the cube root on both the sides of the equation. And then, we have to substitute x in terms of y. Thus, the obtained expression is the required solution, that is, the value of ${{f}^{-1}}\left( x \right)$ when $f\left( x \right)=\dfrac{1}{{{x}^{3}}}$.

Complete answer:
According to the question, we are asked to find ${{f}^{-1}}\left( x \right)$ of the given function $f\left( x \right)=\dfrac{1}{{{x}^{3}}}$.
We have been given the function $f\left( x \right)=\dfrac{1}{{{x}^{3}}}$. -----(1)
We first have to consider $f\left( x \right)=y$.
We get, $y=\dfrac{1}{{{x}^{3}}}$.
Using the method of cross-multiplying, that is, $a=\dfrac{1}{b}\Rightarrow b=\dfrac{1}{a}$.
We get, ${{x}^{3}}=\dfrac{1}{y}$. --------(2)
Let us now take the cube root on both the sides of the equation (2).
$\Rightarrow \sqrt[3]{{{x}^{3}}}=\sqrt[3]{\dfrac{1}{y}}$
$\Rightarrow \sqrt[3]{{{x}^{3}}}=\dfrac{\sqrt[3]{1}}{\sqrt[3]{y}}$
We know that $\sqrt[3]{{{x}^{3}}}=x$ .
And any power raised to 1 is 1.
We get,
$\Rightarrow x=\dfrac{1}{\sqrt[3]{y}}$
But we also know that $\sqrt[3]{y}={{y}^{\dfrac{1}{3}}}$.
Therefore, $x=\dfrac{1}{{{y}^{\dfrac{1}{3}}}}$.
We have now obtained the value of x in terms of y.
Now, to find ${{f}^{-1}}\left( x \right)$ we have to replace y with x.
${{f}^{-1}}\left( x \right)$ is nothing but value of x in terms of x in the given function $f\left( x \right)=\dfrac{1}{{{x}^{3}}}$.
${{f}^{-1}}\left( x \right)=\dfrac{1}{{{x}^{^{\dfrac{1}{3}}}}}$
Hence, the value of ${{f}^{-1}}\left( x \right)$ for the function $f\left( x \right)=\dfrac{1}{{{x}^{3}}}$ is $\dfrac{1}{{{x}^{^{\dfrac{1}{3}}}}}$.

Note: Whenever you get this type of problem, we should always try to make the necessary calculations in the given equation to get the final of x in terms of x which will be the required answer. We should avoid calculation mistakes based on sign conventions. The final solution can also be written as ${{f}^{-1}}\left( x \right)=\dfrac{1}{\sqrt[3]{x}}$.
We can check the final answer by this method: $f\left( {{f}^{-1}}\left( x \right) \right)=x$
Here ${{f}^{-1}}\left( x \right)=\dfrac{1}{{{x}^{^{\dfrac{1}{3}}}}}$.
Therefore, $f\left( \dfrac{1}{{{x}^{^{\dfrac{1}{3}}}}} \right)=\dfrac{1}{{{\left( \dfrac{1}{{{x}^{^{\dfrac{1}{3}}}}} \right)}^{3}}}$ [since $f\left( x \right)=\dfrac{1}{{{x}^{3}}}$]
$\Rightarrow f\left( \dfrac{1}{{{x}^{^{\dfrac{1}{3}}}}} \right)=\dfrac{1}{\left( \dfrac{1}{{{x}^{\dfrac{1}{3}\times 3}}} \right)}$
$\Rightarrow f\left( \dfrac{1}{{{x}^{^{\dfrac{1}{3}}}}} \right)=\dfrac{1}{\left( \dfrac{1}{x} \right)}$
$\Rightarrow f\left( \dfrac{1}{{{x}^{^{\dfrac{1}{3}}}}} \right)=x$
$\therefore f\left( {{f}^{-1}}\left( x \right) \right)=x$
Hence, the obtained answer is verified.