Question

Find the inverse of the function \[y = 2{x^2} - 3x + 1\] and find out whether is it a function or not?

Answer 1

Hint: To find the inverse of an equation then you have to find the value of the second variable in respect of the first variable then the equation obtained is the inverse of the first variable in respect to second. What you need to do is just rearrange the equation so to obtain the desired equation you need to get.

Complete step-by-step answer:
The given equation is \[y = 2{x^2} - 3x + 1\]
Here we have to rearrange the term such that the variable “x” get its equation with respect to the variable “y” on solving we get:
 \[
   \Rightarrow y = 2{x^2} - 3x + 1 \\
   \Rightarrow 2{x^2} - 3x + 1 - y = 0 \\
   \Rightarrow 2{x^2} - 3x + (1 - y) = 0 \;
 \]
From here we are not able to rearrange completely for the variable “x” so we have to find the roots of the formed quadratic equation by using the sridharacharya rule of finding the roots of the quadratic equation, on solving we get:
 \[
   \Rightarrow x = \dfrac{{ - ( - 3) \pm \sqrt {9 - 4 \times 2 \times (1 - y)} }}{4} \\
   \Rightarrow x = \dfrac{{3 \pm \sqrt {1 + 8y} }}{4} \;
 \]
Here we get the inverse of the given equation, in the question.
This is not a function because for one value of “y” we have two value of “x”, one having the plus sign and the other is having the negative sign.
So, the correct answer is “$ x = \dfrac{{3 \pm \sqrt {1 + 8y} }}{4}$”.

Note: The inverse can also be find out by only rearranging the equation, for the variable we are searching for but here for this question because it was a quadratic equation and having the linear term also for which we are finding the inverse, that is why we are here not able to find the inverse