1. Warm Ups! Find f(g(x)) and g(f(x)) for each of the following: 1.F(x)= 2x +1, g(x) = (x-1)/2 2.F(x) = ½ x + 3, g(x) = 2x-6

2. Inverse Functions The secret to solving complicated expressions

3. What Is An Inverse? A function and its inverse “undo” one another, so that the variable is isolated

4. Details, Details Domain: –The domain of the function is the range of the inverse Range: –The range of the function is the domain of the inverse

5. “Creating” the Inverse Methods : 1.Switch the range and domain value for every point in the function 2.Switch the x (or independent variable) and y (or dependent variable) and solve for y (or dependent variable) Use: 1.When function is represented as a set of ordered pairs 2.When function is in equation form

6. Example of Ordered Pairs: F(x) = { (3,1), (-3, 5), (9,2)} What is the inverse? F -1 (x) = {(1,3),(5,-3),(2,9)}

7. Example of Equations 1.F(x) = 3x 2 +1 2.What is the inverse? 3.Y = 3x 2 +1 4.X = 3y 2 +1 5.X-1 = 3y 2 6.(x-1)/3 = y 2 7. = y 8.Is this truly a function? Must restrict the domain, y >0 Why? Original range was only positive numbers

8. How Can I Tell If I’ve Found the Correct Inverse? Graphically: A function and its inverse are reflections in the line y = x Algebraically: If f(f -1 (x)) = x AND f -1 (f(x)) = x THEN F -1 (x) and f(x) are inverses

9. Graphically: A function and its inverse are reflections in the line y = x

10. Algebraically: If f(f -1 (x))) = x AND f -1 (f(x)) = x THEN F -1 (x) and f(x) are inverses 1. 1.F(x)=3x 2 +1 2. 2.f -1 (x) = 3. 3.F(f -1 (x)) = 3( ) 2 +1 3((x-1/3) +1 x-1 +1 x Now you check if f -1 (f(x)) = x

11. f -1 (f(x)) = x 1.F(x)= 3x 2 +1 2.f -1 (x) = 3. f -1 (3x 2 +1) = 4. f -1 (3x 2 +1) = 5.f -1 (3x 2 +1) = 6.f -1 (3x 2 +1) = x 7.Since f -1 (f(x)) = x and f(f -1 (x))) = x, f and f -1 are inverses