Question

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

Answer 1

Hint: 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.