Question

What is the inverse function of $y=2x-1$?

Answer 1

Hint: To obtain the inverse function of the equation we will use a method to find the inverse of a linear function. Firstly we will switch the $x$ and $y$ in the equation and then we will simplify the equation to find the value of the new $y$ by taking it on one side and the other terms on the other side and get our desired answer which is the inverse function.

Complete step by step solution:
The Linear function is given as below:
$y=2x-1$
First step will be to switch value of $x$ and $y$ in above equation as follows:
$x=2y-1$
Next we will add 1 on both sides of the equation and simplify as below:
$\begin{align}
  & \Rightarrow x+1=2y-1+1 \\
 & \Rightarrow x+1=2y \\
\end{align}$
Now divide both sides by 2 and simplify as below:
$\begin{align}
  & \Rightarrow \dfrac{x+1}{2}=\dfrac{2y}{2} \\
 & \therefore y=\dfrac{x+1}{2} \\
\end{align}$
We can write the above inverse as:
${{f}^{-1}}\left( x \right)=\dfrac{x+1}{2}$

Hence we get the inverse of $y=2x-1$ as ${{f}^{-1}}\left( y \right)=\dfrac{x+1}{2}$

Note: A function is any rule or a mechanism which is defined to give a relationship between the elements of the two sets. There are various types of function such as into function onto function and equal function etc. To check whether a function has an inverse function we use Horizontal line Test. An inverse function or an anti-function is a function that reverses another function. A function has an inverse if and only if for every $y$ in the range there is only one $x$ in its domain. Then the inverse function associates every element in its range to only one element in its range. The inverse is unique of a function if it is both one-one and onto.