Algebraically finding inverses

The Algebraically finding inverses exercise appears under the Algebra II Math Mission and Mathematics III Math Mission. This exercise finds the inverses of functions by an algebraic process.

Types of Problems

There is one type of problem in this exercise:

1. Find the inverse of the function: This problem provides a function that has an inverse. The student is expected to take the function provided, find its inverse, and write it in the space provided.

   

   Find the inverse of the function

Strategies

Knowledge and comfort with the process of finding inverses and what they represent are encouraged to ensure success on this exercise.

1. A mechanical process to find inverses:
   1. Call ${f(x)=y}$
   2. Switch ${x}$ and ${y}$
   3. Solve for ${y}$
   4. Call ${y=f^{-1}(x)}$
2. The inverse of a rational function tends to be another rational expression. Solving for this requires cross multiplication and distributive properties.
3. The inverse of a radical function tends to be a polynomial-like function.

Real-life Applications

1. A function has much more usefulness if it can be undone to find inputs instead of outputs. That is what inverse functions do (where possible).
2. Inverse functions are useful for finding range of functions, as the range of a function is the domain of it's inverse (if it exists).