1. 6.4 Inverse Functions Part 1 Goal: Find inverses of linear functions.

2. Inverse Functions A function and its inverse can be described as the "DO" and the "UNDO" functions. A function and its inverse can be described as the "DO" and the "UNDO" functions. A function takes a starting value, performs some operation on this value, and creates an output answer. A function takes a starting value, performs some operation on this value, and creates an output answer. The inverse of this function takes the output answer, performs some operation on it, and arrives back at the original function's starting value.

3. Definition of an Inverse The inverse of a function is the set of ordered pairs obtained by interchanging the first and second elements of each pair in the original function. The inverse of a function is the set of ordered pairs obtained by interchanging the first and second elements of each pair in the original function. Notation: If f (x) is a given function, then f -1 (x) denotes the inverse of f.

4. Finding the Inverse of a Function Basically, the process of finding an inverse is simply the swapping of the x and y coordinates. Basically, the process of finding an inverse is simply the swapping of the x and y coordinates. This newly formed inverse will be a relation, but may not necessarily be a function. This newly formed inverse will be a relation, but may not necessarily be a function.

5. Horizontal Line Test The inverse of a function f is also a function if and only if no horizontal line intersects the graph of f more that once.

6. Three Methods to find the Inverse: 1.Reflect graph over the line y = x. 2.Swapping x and y-values 3.Solving Algebraically: a. Set the function = y b. Swap the x and y variables c. Solve for y

7. Reflect the graph over y = x Graph original function f(x) = 2x + 3 It is drawn in blue. If reflected over the identity line, y = x, the original function becomes the red dotted graph. dotted graph.

8. Swapping x and y-values Given relation, find the inverse relation. x-2012 y420-2-4 x420-2-4y-2012

9. Solving Algebraically Find the equation of the inverse of the relation f(x) = 2x – 4. 1.Set the function = y y = 2x – 4 2.Swap the x and y variables x = 2y -4 3.Solve for y y = (x + 4)/2

10. Find the equation for the Inverse function. f(x) = -2x + 5 f(x) = -2x + 5 f -1 (x) = -(x – 5)/2

11. Verifying Inverse Functions Verify that f(x) = 2x – 4 and f -1 (x) = ½x + 2 are inverses.

12. 6.3 Composition Functions (Inverse Functions) Part 2 Goal: Evaluate composition functions and prove functions are inverses of each other using composition functions.

13. Composition of Functions The composition of a function g with a function f is: h(x) = g(f(x))

14. Example 1 Let f(x) = 3x – 14 and g(x) = x + 5. What is the value of g(f(4))? First find f(4).First find f(4). f(x) = 3x – 14 f(4) = 3(4) – 14 = -2 Then find g(-2).Then find g(-2). g(x) = x + 5 g(-2) = -2 + 5 = 3

15. Example 2 Let f(x) = 3x + 2 and g(x) = 2x – 7. Find f(g(x)). Find g(x) first.Find g(x) first. g(x) = 2x – 7 Then find f(2x – 7)Then find f(2x – 7) f(x) = 3x + 2 f(2x – 7) = 3(2x – 7) + 2 f(2x – 7) = 6x – 21 + 2

16. f(2x – 7) = 6x – 19 Therefore, f(g(x)) = 6x – 19

17. This "DO" and "UNDO" process can be stated as a composition of functions. If functions f and g are inverse functions, f(g(x)) = g(f(x)) = x. Example: If f(x) = x-1 and g(x) = x +1 Example: If f(x) = x-1 and g(x) = x +1 then f(g(x)) = x and g(f(x)) = x Think of them as "undoing" one another and leaving you right where you started.

18. Verify that f and g are inverse functions. f(x) = 3x – 1 and g(x) = ⅓x + ⅓ f(x) = 3x – 1 and g(x) = ⅓x + ⅓