Question

How do you find \[{{f}^{-1}}\left( x \right)\] given \[f\left( x \right)=\dfrac{1}{x+2}\]?

Answer 1

Hint: Assume the given function \[f\left( x \right)\] equal to y. Now, cross – multiply the terms and find the value of x in terms of y, that means x as a function of y. Assume the obtained expression as equation (i). Now, use the substitution that if \[f\left( x \right)=y\] then \[x={{f}^{-1}}\left( y \right)\]. Consider it as equation (ii) and equate the two expressions. Replace y with x to get the expression for \[{{f}^{-1}}\left( x \right)\].

Complete step by step answer:
Here, we have been provided with the function \[f\left( x \right)=\dfrac{1}{x+2}\] and we are asked to find its inverse function. That means we have to find the expression of \[{{f}^{-1}}\left( x \right)\].
First of all note that the given function will be undefined for x = -2 because in that case the denominator of \[f\left( x \right)\] will become 0. So, \[x\ne -2\].
Now, assuming the given function \[f\left( x \right)\] equal to y, we get,
\[\Rightarrow y=\dfrac{1}{x+2}\]
By cross – multiplying, we get,
\[\Rightarrow xy+2y=1\]
Now, we have to find x in terms of y, that means x as a function of y. So, let us rearrange the terms to make the coefficient of x equal to 1.
\[\Rightarrow xy=1-2y\]
Dividing both the sides with y, we get,
\[\Rightarrow x=\dfrac{1-2y}{y}\] - (1)
Here, y must not be 0, i.e., \[y\ne 0\], otherwise the function will become undefined.
You may recall that we have assumed \[f\left( x \right)=y\], so we can write: -
\[\Rightarrow x={{f}^{-1}}\left( y \right)\] - (2)
From equations (1) and (2), we get,
\[\begin{align}
  & \Rightarrow \dfrac{1-2y}{y}={{f}^{-1}}\left( y \right) \\
 & \Rightarrow {{f}^{-1}}\left( y \right)=\dfrac{1-2y}{y} \\
\end{align}\]
Replacing the variable y with the variable x on both the sides, we get,
\[\Rightarrow {{f}^{-1}}\left( x \right)=\dfrac{1-2x}{x}\]

Hence, the above relation represents the inverse of the given function \[f\left( x \right)\].

Note: One may note that you may simplify the expression \[\dfrac{1-2x}{x}\] further by breaking the terms. It will be given as: - \[\dfrac{1-2x}{x}=\dfrac{1}{x}-2\]. You must check the values of x for which the function is undefined. Always remember that the denominator of the function cannot be zero. After finding the expression \[{{f}^{-1}}\left( y \right)\] do not forget to replace y with x because every time we will be asked to find \[{{f}^{-1}}\left( x \right)\] and not \[{{f}^{-1}}\left( y \right)\]. Remember the steps that we have followed to find the inverse function because there is no easier method apart from the above method. Do not get confused and consider the reciprocal of \[f\left( x \right)\] as the answer as it will be the wrong approach.