1. 1 11-5-15 AM5.2c To Find the Inverse of a Function One cheesy tortilla chip…. Two cheesy tortilla chips…. Three cheesy tortilla chips…. Five cheesy tortilla chips…. Eight cheesy tortilla chips…. Math geeks shouldn’t be allowed anywhere near certain foods. What’s wrong with fibonachos? There’s only 12 left. Now what? Got ID?

2. 2 Opener & Lesson—Copy this page. Definition: One to one function: A function (x’s don’t repeat and passes the vertical line test) in which the y’s don’t repeat and passes the horizontal line test. This is important in order for a function to have an inverse. Given one function, f(x), it’s inverse is f -1 (x) (reads “f inverse of x”) and they switch domains and ranges. In other words, the x’s of the first become y’s of the second and the y’s of the first become x’s of the second. A function has an inverse if and only if it is one to one.

3. 3 Active Learning Assignment? Let’s look at the graphs of a function and it’s inverse

4. 4 The green dotted line represents the axis of symmetry y = 2x y = ½ x (2, 4) (4, 2) y = x

5. 5 …is not a one to one function because it fails the horizontal line test. But, half of it is!

6. 6 Ex: We want to find the inverse for: But, let’s look at the graph, first. And it passes both the horizontal and the vertical line tests!

7. 7 1. Change f(x) to y 2. Switch out x and y 3. Solve for y Look at the graph, again. Now, find the inverse for: 4. Change y to f -1 (x)

8. 8 But, when you put them together: (4,0) (0,4)

9. 9 Find f -1 (x) of: f(x) = 2x + 3 y = 2x + 3 x = 2y + 3 x – 3 = 2y Important: If you have an equation with an x 2 or any even integer power or |x| term, either one with no restrictions, the function does not have an inverse. (Fails the horizontal line test!) 1. Change f(x) to y 2. Switch out x and y 3. Solve for y 4. Change y to f -1 (x) y = x 2 y = |x|

10. 10 Now, fyi, let’s look at these two functions as composites: f(x) = 2x + 3 and They inverse each other out!

11. 11 Find f -1 (x) of: f(x) = -3x – 7 y = -3x – 7 1. Change f(x) to y x = -3y – 7 x + 7 = -3y 2. Switch out x and y 3. Solve for y 4. Change y to f -1 (x) Look at page 150: #11-19. Discuss with your group as to which ones have inverses and explain. (Hint: if they have |x| or x 2 or any even integer power, then the answer is “NO”.

12. 12 Active Learning Assignment: P. 150: 11-19 (Don’t worry about finding the rule.) Remember: If you have an equation with an x 2 or any even integer power or |x| term, either one with no restrictions, the answer is “NO”, and you don’t have to do anything else. Test on Domain & Range, Part 2: Tues., 11/10