Summary

• For a function to have an inverse, the function must be one-to-one
  • Given the graph of a function, we can determine whether the function is one-to-one by using the horizontal line test
• If a function is not one-toone, we can restrict the domain where the function becomes one-to-one
  • This way, the new, restricted domain function can be inverted
• For a function $f$ and its inverse $f^{-1}$, $f(f^{-1}(x)) = x$ in the domain of $f^{-1}$ and $f(f^{-1}(x)) = x$ for all $x$ in the domain of $f$
• Since the trigonometric fucntions are periodic, we need to restrict their domains to define the inverse trigonometric functions
• The graph of a function $F$ and its inverse $f^{-q}$ are symmetric about the line $y = x$

Let $f(x)$ be a function that is defined for all $x$ on some interval $I$. If, for each value of $y$ in the range of $f$ there is one (and only one), value of $x$ in the interval $I$ such that $f(x) = y$, then we define an inverse function $f^{-1}$ according to the following rule:

<aside> ⚡ $f^{-1}(y)$ is the unique $x$ in the interval $I$ such that $y = f(x)$

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How to determine if a function has an inverse?

How to find $f^{-1}(x)$?

Inverse Trigonomic Functions