Composite Function: Combining a function within another function. Notation: Function “f” of Function “g” of x “x to function g and then g(x) into function f” Example 1 Evaluate Composites of Functions Recall: (a + b)2 = a2 + 2ab + b2 a) b)

Example 2 Composites of a Function Set g(x) f(x) f(g(x)) X Y 3 5 7 9 11 X Y 7 3 5 9 8 11 4 X Y 3 5 7 8 9 4 This means f(g(5))=3

Example 2 Composites of a Function Set b) f(x) g(x) X Y 7 3 5 9 8 11 4 X Y 5 7 3 9 11 In set form, not every x-value of a composite function is defined

Evaluate Composition Functions f(g(3)) b) g(f(-1))

Evaluate Composition Functions c) d)

Inverse Functions and Relations Inverse Relation: Relation (function) where you switch the domain and range values Function  Inverse Function  Inverse Domain of the function  Range of Inverse and Range of Function  Domain of Inverse Inverse Notation: Inverse Properties: 1] Input a into function and output b, then inverse function will input b and output a (switch) 2] Composition of function and inverse or vice versa will always equal x (original input)

Steps to Find Inverses One-to-One: [1] Replace f(x) with y ONE TO ONE: There are no repeated x or y values. Every x is paired with exactly one unique (DIFFERENT) y-value [1] Replace f(x) with y [2] Interchange x and y [3] Solve for y and replace it with One-to-One: A function whose inverse is also a function (horizontal line test) Function Inverse Inverse is not a function

Example 1: Inverses of Ordered Pair Relations b) c) Are inverses f-1(x) or g-1(x) functions? f-1(x) not a function because x repeats

Inverses of Graphed Relations FACT: The graphs of inverses are reflections about the line y = x Find inverse of y = 3x - 2 y = 1/3x + 2/3 x = 3y – 2 x + 2 = 3y 1/3x + 2/3 = y y= x y= 3x - 2

Example 3 Find an Inverse Function b)

Example 3 Continued c) d) PART D) Function is not a 1-1. (see example) So the inverse is 2 different functions: If you restrict the domain in the original function, then the inverse will become a function. (x > 0 or x < 0)

Example 4: Verify two Functions are Inverses Method 1: Directly solve for inverse and check Method 2: Composition Property Yes, Inverses Yes, Inverses