Question

What would the inverse of $y=\dfrac{1}{x}$ be?

Answer 1

We first explain the expression of the function. We convert the function from $y$ of $x$ to $x$ of $y$. The inverse function on being conjugated gives the value of $x$. At the end we interchange the terms to make it a general equation.

Complete step by step solution:
Let us take an arbitrary number $m$. The reciprocal of the number $m$ is $z$ then we have $mz=1$ which gives $z=\dfrac{1}{m}$.
We need to find the inverse of the equation of $y=\dfrac{1}{x}$.
The given equation is a function of $x$ where we can write $y=f\left( x \right)$.
If we take the inverse of the equation, we will get $x={{f}^{-1}}\left( y \right)$.
The given function was of $x$. We convert it to a function of $y$ and that becomes the inverse.
We need to express the value of $x$ with respect to $y$.
From the condition we get $y=\dfrac{1}{x}$ which gives $xy=1$. Therefore, $x=\dfrac{1}{y}$
From $x=\dfrac{1}{y}$, we get $y=\dfrac{1}{x}$. So, $y={{f}^{-1}}\left( x \right)=\dfrac{1}{x}$

Therefore, the inverse function of $y=\dfrac{1}{x}$ is itself.

Note: We can verify the result by taking the composite function. We have two functions being inverse to each other. They are $f\left( x \right)=\dfrac{1}{x}$ and ${{f}^{-1}}\left( x \right)=\dfrac{1}{x}$.
If we take $f\left( {{f}^{-1}}\left( x \right) \right)$, we will get $x$.
So, $f\left( {{f}^{-1}}\left( x \right) \right)=f\left( \dfrac{1}{x} \right)=\left( \dfrac{1}{\dfrac{1}{x}} \right)=x$.
Thus, the inverse function of $y=\dfrac{1}{x}$ is itself.