Calculus – Inverse functions

Definition: A function \(f\) is called one-to-one if

\[f(x_1)\neq f(x_2)\quad\text{for all}\quad x_1\neq x_2.\]

Definition: If \(f\) is a one-to-one function with domain \(A\) and range \(B\), then we have: the inverse function \(f^{-1}\) has domain \(B\) and range \(A\), and

\[f^{-1}(y)=x\quad\Longleftrightarrow\quad y=f(x).\]

How to find the inverse of a function?

The inverse of a one-to-one function can be obtained as follows:

• Step 1: start with \(y=f(x)\).
• Step 2: solve this equation for \(x\) (in terms of \(y\)).
• Step 3: interchange the role of \(x\) and \(y\), then we have: \(y=f^{-1}(x)\).

The graphs of \(f\) and \(f^{-1}\) are each other's reflection in the line \(y=x\).

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Last modified on March 1, 2021