Question

Find a formula for the inverse of the function\(y = {x^2} - x,x \ge \frac{1}{2}\).

Short answer

The formula for the inverse of the function \(y = {x^2} - x,x \ge \frac{1}{2}\) is \({f^{ - 1}}(x) = \frac{1}{2} + \sqrt {x + \frac{1}{4}} \).

Step-by-step solution

Given data

The given function is \(y = {x^2} - x\).

Concept of functions

The simplest definition is an equation will be a function if, for any \({\rm{x}}\) in the domain of the equation (the domain is all the \({\rm{x}}\)'s that can be plugged into the equation), the equation will yield exactly one value of \({\rm{y}}\) when we evaluate the equation at a specific \({\rm{X}}\).

Solve the equation

Solve this equation for \(x\) as shown below.

\(\begin{array}{c}y + {\left( {\frac{1}{2}} \right)^2} = {x^2} - x + {\left( {\frac{1}{2}} \right)^2}\\y + \frac{1}{4} = {\left( {x - \frac{1}{2}} \right)^2}\\\sqrt {y + \frac{1}{4}} = \pm \left( {x - \frac{1}{2}} \right)\end{array}\)

Apply completing the square method.

Notice that the \(x\) is restricted as \(x \ge \frac{1}{2}\) and hence \(x - \frac{1}{2} \ge 0\).

Thus, ignore the negative root.

Then, \(\sqrt {y + \frac{1}{4}} = x - \frac{1}{2}\) from which the value of \(x = \frac{1}{2} + \sqrt {y + \frac{1}{4}} \).

Then, interchange \(x\) and \(y\), obtain the inverse function, \(y = \frac{1}{2} + \sqrt {x + \frac{1}{4}} \).

Thus, the required inverse function is \({f^{ - 1}}(x) = \frac{1}{2} + \sqrt {x + \frac{1}{4}} \).