Question

How do you find the inverse of \[y = {x^2} - 2x + 1\] and is it a function?

Answer 1

Hint: Here in this question we have to find the inverse of the function, if the function is of the form exponential number we use the concept of logarithm function and on further simplification we obtain the required solution for the given question. To find the inverse we swap the terms.

Complete step by step solution:
In mathematics, an inverse function is a function that "reverses" another function: if the function \[f\] applied to an input x gives a result of y, then applying its inverse function \[g\] to y gives the result x, i.e., \[g(y) = x\] if and only if \[f(x) = y\]. The inverse function of \[f\] is also denoted as \[{f^{ - 1}}\]
Now consider the given equation
\[y = {x^2} - 2x + 1\]
The RHS of the equation is in the form of \[{a^2} - 2ab + {b^2}\], we have standard formula for this algebraic identity \[{a^2} - 2ab + {b^2} = {(a - b)^2}\]. When we compare the given RHS term it is similar to the standard algebraic equation \[{a^2} - 2ab + {b^2} = {(a - b)^2}\]. Therefore, the given equation can be written as
\[ \Rightarrow y = {(x - 1)^2}\]
Taking the square root we have
\[ \Rightarrow \sqrt y = x - 1\]
Take -1 to the LHS, while taking -1 to the LHS the sign will get change.
\[ \Rightarrow \sqrt y + 1 = x\]
Hence we have determined in the terms of x.
Now replace the term x by term y and vice versa.
\[ \Rightarrow y = \sqrt x + 1\]
When the given equation is written in the form of the function it is given as
\[f(x) = {x^2} - 2x + 1\]
The inverse function is given as
\[ \Rightarrow {f^{ - 1}}(x) = \sqrt x + 1\]
So, the correct answer is “\[ {f^{ - 1}}(x) = \sqrt x + 1\]”.

Note: We must know about the simple arithmetic operations. To find the inverse we swap the y variable into x and simplify the equation and determine the value for y. since the solution we obtain will be in the form of square root on simplification. While shifting the terms we must take care of signs.