Question

Find the inverse of the cube root function f.
\[f(x)={{(x+1)}^{1/3}}\]
\[\begin{align}
  & \left( A \right)\text{ }{{\text{x}}^{3}}+1 \\
 & \left( B \right)\text{ }{{\text{x}}^{3}}-1 \\
 & \left( C \right)\text{ x-1} \\
 & \left( D \right)\text{ x+1} \\
\end{align}\]

Answer 1

Hint: Let us assume \[f(x)={{(x+1)}^{1/3}}\] as equation (1). Now let us assume \[f(x)=y\]. Now let us consider this equation as equation (2). Now we should substitute equation (2) in equation (1). Now we should do cubing on both sides. Let us consider this as equation (4). Now we should substitute equation (2) in equation (4). Now let us consider this equation as equation (5). We know that we should reverse the inverse function. So, let us substitute \[x={{f}^{-1}}(y)\]. Let us assume this as equation (6). Now we should replace y with x. This will give us the inverse function of \[f(x)={{(x+1)}^{1/3}}\].

Complete step-by-step answer:
From the question, it was given that \[f(x)={{(x+1)}^{1/3}}\].
Let us assume
\[f(x)={{(x+1)}^{1/3}}..........(1)\]
Let us assume
\[f(x)=y........(2)\]
Now we should substitute equation (2) in equation (1), we get
\[y={{(x+1)}^{1/3}}..........(3)\]
Now we should do cubing on both sides, we get
\[\begin{align}
  & \Rightarrow {{y}^{3}}={{\left( {{\left( x+1 \right)}^{1/3}} \right)}^{3}} \\
 & \Rightarrow {{y}^{3}}=x+1 \\
 & \Rightarrow x={{y}^{3}}-1......(4) \\
\end{align}\]
Now we should substitute equation (2) in equation (4), we get
\[\Rightarrow x={{(f(x))}^{3}}-1.......(5)\]
From the question, we have to find the inverse function of f(x).
Now we substitute\[x={{f}^{-1}}(y)\].
So, from equation (2), we get
\[\begin{align}
  & \Rightarrow {{f}^{-1}}f(x)={{f}^{-1}}(y) \\
 & \Rightarrow x={{f}^{-1}}(y)......(6) \\
\end{align}\]
Now we should substitute equation (6) and equation (2) in equation (5), we get
\[\Rightarrow {{f}^{-1}}(y)={{y}^{3}}-1.....(7)\]
Now we will replace y with x in equation (7).
\[\Rightarrow {{f}^{-1}}(x)={{x}^{3}}-1.....(8)\]
Hence, option B is correct.

Note:In this question, if we do not replace y with x then we will get the inverse function in terms of y. Then we cannot find the correct option. So, it is required to replace y with x to find the correct option. Students should be careful at calculation part of this question. If a small mistake is made, then it is not possible to get a correct final answer. So, one should do the calculation part in a perfect manner. This is also important to solve this problem.