Find a formula for the inverse of the function. y=x^2-x, x...

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Key Concepts

Inverse Function

An inverse function reverses the effect of the original function, mapping outputs back to inputs. Finding an inverse involves expressing the independent variable in terms of the dependent variable so that applying the inverse function will yield the original input when the function is applied.

One-to-One Functions and Domain Restrictions

A function must be one-to-one (injective) for an inverse to exist, meaning each output is produced by exactly one input. Domain restrictions are often applied to functions, especially quadratic ones, to ensure one-to-one behavior so that the process of finding an inverse is valid.

Quadratic Equation

A quadratic equation is a polynomial equation of degree two, and solving it often involves rewriting the equation and applying methods such as the quadratic formula. In the context of finding inverses, the resulting equation in the independent variable is quadratic when the original function is quadratic.

Quadratic Formula

The quadratic formula provides a method for finding the roots of any quadratic equation ax² + bx + c = 0. It is given by x = (-b ± sqrt(b² - 4ac)) / (2a), and is essential when solving for the inverse of a quadratic function to determine the correct branch that adheres to the imposed domain restrictions.

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