Question

How do you find the inverse of $ y = \dfrac{1}{x} $ and is it a function?

Answer 1

Hint: Here we will take the given function and will find the inverse of the function. We are given an equation containing the fraction on one side which is expressed as the ratio of the number expressed in the form of numerator to denominator.

Complete step-by-step answer:
Take the given expression –
 $ y = \dfrac{1}{x} $
The above equation can be re-written as –
 $ \dfrac{y}{1} = \dfrac{1}{x} $
To find the inverse of the above expression, replace the numerator with the value of the denominator and replace the denominator with the numerator's value.
 $ \Rightarrow \dfrac{1}{y} = \dfrac{x}{1} $
The above equation can be re-written as –
 $ \Rightarrow \dfrac{1}{y} = x $
Move left hand side value on the right hand side and vice-versa.
 $ \Rightarrow x = \dfrac{1}{y} $
The inverse of the given function is also the function
So, the correct answer is “ $ x = \dfrac{1}{y} $ ”.

Note: Reciprocal is also known as the “multiplicative inverse” which is simply one of a pair of numbers that, when multiplied together is equal to one. A rational number is the number which can be expressed as the ratio of two numbers or which can be expressed as the p/q form or as the quotient or the fraction with non-zero denominator whereas, the numbers which are not represented as the rational are known as the irrational number.
Let us take example for the rational number such as –
 $ \dfrac{3}{4} $
so the reciprocal will be $ \dfrac{4}{3} $
Where the numerator goes to the denominator and the denominator goes to the numerator.