- Use composition to check if two functions are inverses
- Determine the domain and range of an inverse function, and restrict the domain of a function to make it one-to-one
- Find or calculate the inverse of a function
- Use the graph of a function to draw its inverse
Shoe Size Conversions
Maya works at an international shoe store and constantly needs to convert between US women’s sizes and European sizes. She knows the conversion pattern:
- US size 5 → European size 35
- US size 6 → European size 36
- US size 7 → European size 37
- US size 8 → European size 38
- US size 9 → European size 39
Maya notices that to convert from US to European, she adds 30: [latex]f(x)=x+30[/latex] where [latex]x[/latex] is the US size.
But customers also ask “What US size is European 37?” Now Maya needs to work backwards. She needs the inverse function.
An inverse function, written as [latex]f^{-1}(x)[/latex], reverses what the original function [latex]f(x)[/latex] does.
Properties: [latex]f(f^{-1}(x)) = x[/latex] and [latex]f^{-1}(f(x)) = x[/latex]
For Maya’s shoe conversions:
- [latex]f(x) = x + 30[/latex] converts US to European
- [latex]f^{-1}(x) = x - 30[/latex] converts European to US