1. Algebra 2 Unit 9: Functional Relationships
Topic: Functions & Their Inverses

2. Vocabulary Inverse Relation One-to-one Function
A relation that “undoes” a function.
The domain of a function is the range of its inverse; the range of a function is the domain of its inverse.
The graphs of a function & its inverse are symmetric about the line y = x.
One-to-one Function
A function in which each range value is paired with one and only one domain value.
If a function, f is one-to-one, then it’s inverse is also a one-to-one function and is notated f -1.

3. Determining whether a function is one-to-one
If a function passes the horizontal line test, it is a one-to-one function.
Any horizontal line must pass through the graph of a function once and only once.
Function is one-to-one. Inverse will also be a function.
Function is not one-to one. Inverse will not be a function.

4. Finding the inverse of a function
Replace f (x) with y, then switch x & y in the equation.
Solve the resulting equation for y.
Take the square root of both sides (remember there are two solutions).
Subtract 2 from both sides.
Multiply both sides by 2.
The resulting relation is the inverse of f (x).

5. Inverse Functions
Determine if the given function is one-to-one. If so, find its inverse & state its domain & range.
The graph of the function does not pass the horizontal line test. It is not one-to-one, therefore its inverse is not a function (and we’re done with this problem!)

6. Inverse Functions
Determine if the given function is one-to-one. If so, find its inverse & state its domain & range.
The graph of the function does pass the horizontal line test. It is one-to-one, therefore its inverse is a function, and we must find it.

7. Inverse Functions
Determine if the given function is one-to-one. If so, find its inverse & state its domain & range.
Replace f (x) with y and switch x & y.
Solve for y to find the inverse.
Since we know this is a function, we must notate it properly (change y to f -1).
The domain & range of f (x) is all real #s, thus the domain & range of f -1(x) is all real #s.

8. Quest: Functions & Their Inverses Due 5/7 (A-day) or 5/8 (B-day)
Homework
Quest: Functions & Their Inverses
Due 5/7 (A-day) or 5/8 (B-day)