Question

How do you find the inverse of \[f(x)=1-\left( \dfrac{1}{x} \right)\]?

Answer 1

Hint: An inverse function is defined as a function, which can reverse into another function, or if a function takes \[x\] to \[y\] then the inverse function takes \[y\] to \[x\]. If there is a function \[f(x)\] having an inverse function \[g(x)\]. Then the domain of the function \[f(x)\] is the range of the function \[g(x)\] and the domain of \[g(x)\] is the range of the function \[f(x)\]. The inverse of a function is represented as \[{{f}^{-1}}(x)\], one shouldn’t confuse the -1 with the exponent here. Once we know that the inverse function exists, it can be found by following the steps below
1. First, replace \[f(x)\] with \[y\].
2. Replace every \[x\] with a \[y\] and replace every \[y\] with an \[x\] .
3. Solve the equation from Step 2 for \[y\].
4. Replace \[y\] with \[{{f}^{-1}}(x)\].

Complete step by step answer:
The given function is \[f(x)=1-\left( \dfrac{1}{x} \right)\], we have to find its inverse. To find the inverse of a function, we have to follow a set of certain steps as follow,
We will now these steps accordingly, in the given function \[f(x)=1-\left( \dfrac{1}{x} \right)\], replacing \[f(x)\] with \[y\] we get, \[y=1-\left( \dfrac{1}{x} \right)\]
Replacing every \[x\] with a \[y\] and every \[y\] with an \[x\], we get \[x=1-\left( \dfrac{1}{y} \right)\]
Subtracting \[1\] from both sides of the above equation we get,
\[\Rightarrow x-1=1-\left( \dfrac{1}{y} \right)-1\]
\[\Rightarrow x-1=-\left( \dfrac{1}{y} \right)\]
Multiplying \[\dfrac{y}{(x-1)}\] to both sides of the equation, we get
\[\Rightarrow \left( x-1 \right)\dfrac{y}{(x-1)}=-\left( \dfrac{1}{y} \right)\dfrac{y}{(x-1)}\]
\[\therefore y=\dfrac{-1}{x-1}\]

Finally replacing \[y\] with \[{{f}^{-1}}(x)\] we get, \[{{f}^{-1}}(x)=\dfrac{-1}{x-1}\] is the inverse of the given function.

Note: We can check whether our answer is correct or not by checking the domain and range of function.
For function \[f(x)=1-\left( \dfrac{1}{x} \right)\], Domain \[\in (-\infty ,0)\cup (0,\infty )\] and Range \[\in (-\infty ,1)\cup (1,\infty )\]
And for function \[g(x)=\dfrac{-1}{x-1}\], Domain \[\in (-\infty ,1)\cup (1,\infty )\] and Range \[\in (-\infty ,0)\cup (0,\infty )\]
So as the Domain of one function is the range of the other and the range of one is the domain of the other, our answer is correct.