1. Do Now Evaluate the function:

2. Homework Need help? Look in section 7.7 – Inverse Relations & Functions in your textbook Worksheet: Inverses WS

3. Day 12: Inverses

4. Objective To find the inverse of a relation using: A mapping diagram A table An equation To determine if an inverse relation is a function using: A mapping diagram A table A graph An equation

5. Inverse Relations What is an inverse relation? A relation that “undoes” a relation or function The inverse of any relation is written as: Domain (x – values) Range (y – values) xf(x)f(x) 0 1 2 2 4 6 8 2 4 6 8 0 1 2 Function: Inverse: “inverse of f”

6. Inverse Relations Another way to think about it: Inverses switch the domain & range of a relation DOMAIN RANGE 0 1 2 24682468 Relation: (-1, 2), (0, 4), (1, 6), (2, 8) Inverse: (2, -1), (4, 0), (6, 1), (8, 2) Relation DOMAIN RANGE 0 1 2 24682468 Inverse Is the relation and/or the inverse a function? Yes, all elements in both domains correspond to one element in the range

7. Find the inverse of the given relation. State if the relation and/or its inverse are functions. DOMAIN RANGE 4 0 5 -3 2 -6 9 xy 35 -6-3 21 -6-4 Inverse: xy 53 -3-6 12 -4-6 DOMAIN RANGE 4 0 5 -3 2 -6 9 FUNCTION NOT A FUNCTION FUNCTION

8. Step 3: solve for y Step 2: switch x and y Find the inverse of the function. Step 1: put “y” for “f(x)” Step 4: Write using inverse notation Determine if the inverse is a function. Yes, the inverse is a function because all lines pass the VLT.

9. Inverse Relations as Functions If two equations are inverses, you will be able to apply the function, then its inverse and get back to the original value. Find Both f(g(x)) = x and g(f(x)) = x to be inverses.

10. Find the inverse and determine if it is a function. Determine if the two functions are inverses. Choose any value for x and see if No, the functions are not inverses. Yes, the inverse is a function because all lines are functions. Both f(g(x)) = x and g(f(x)) = x to be inverses.

11. Inverse Relations Function: Inverse: xf(x)f(x) 2 04 16 28 All inverses have a line of symmetry at y = x

12. Determine if the graphs depict functions and their inverses. Yes, the graphs show functions & inverses because there is a line of symmetry at y = x. No, the graphs do not show functions & inverses because there is no line of symmetry at y = x.

13. Properties of Inverses If a relation maps a onto b; then an inverse “undoes” the relation and maps b back onto a The domain of the relation is the range of the inverse. The range of the relation is the domain of the inverse. Graphs of relations & inverses have a line of symmetry at the equation y = x Domain of f Range of f -1 Range of f Domain of f -1 a b f f -1

14. Did you meet today’s objective? What does an inverse do? Describe how to find an inverse when given: A table A mapping diagram A graph An equation