Inverse functions
An inverse function “undoes” a function.
If a function takes an input x and outputs y, then the inverse takes that y and gives you back the original x.
Idea: \( f(x) = y \) and \( f^{-1}(y) = x \)
How to find an inverse
- Write \( y = f(x) \)
- Swap \(x\) and \(y\)
- Solve for \(y\)
- Write the result as \( f^{-1}(x) \)
Example
Start with \( f(x) = 2x + 3 \)
1) \( y = 2x + 3 \)
2) Swap: \( x = 2y + 3 \)
3) Solve: \( x - 3 = 2y \Rightarrow y = \dfrac{x - 3}{2} \)
Inverse: \( f^{-1}(x) = \dfrac{x - 3}{2} \)
Graph tip: A function and its inverse are mirror images across the line \(y=x\).
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