Function A FUNCTION is a mathematical “rule” that for each “input” (x-value) there is one and only one “output” (y – value). Set of Ordered Pairs: (input, output) or (x, y) No x-value is repeated!!! A function has a DOMAIN (input or x-values) and a RANGE (output or y-values) For Graphs, Vertical Line Test: If a vertical line can be drawn anywhere on the graph that it touches two points, then the graph is not a function

Function Representations: f is 2 times a number plus 5 Mapping: Set of Ordered Pairs: {(-4, -3), (-2, 1), (0, 5), (1, 7), (2, 9)} Function Notation: f(-4) = -3f(1) = 7 f(-2) = 1f(2) = 9 f(0) = 5 Graph xy Table

Examples of a Function { (2,3), (4,6), (7,8), (-1,2), (0,4), (-2, 5), (-3, -2)} xy #1: Graphs #2: Table #3: Set #4: Mapping

Non – Examples of a Function {(-1,2), (1,3), (-3, -1), (1, 4), (-4, -2), (2, 0)} x 210 y #1: Graphs #2: Table #3: Set #4: Mapping

Practice: Is it a Function? 1.{(2,3), (-2,4), (3,5), (-1,-1), (2, -5)} 2.{(1,4), (-1,3), (5, 3), (-2,4), (3, 5)} x3235 y-2482 #1: No #2: Yes #3: No #4: No #5: No #6: Yes

Function Notation Function Notation just lets us see what the “INPUT” value is for a function. (Substitution Statement) It also names the function for us – most of the time we use f, g, or h. Examples: f(x) = 2x Reads as “f of x is 2 times x” f(3) = 2 * (3) = 6 The (3) replaces every x in rule for the input. Examples: g(x) = 3x 2 – 7x Reads as “g of x is 3 times x squared minus 7 times x ” g(-1) = 3(-1) 2 – 7(-1) = 10 The (-1) replaces every x in rule for the input.

Given f: a number multiplied by 3 minus 5 f(x) = 3x – 5 2) Find f(2)3) Find f(3x)1) Find f(-4) 5) Find f(x) + f(2)4) Find f(x + 2) = 3( x+ 2) – 5 = 3x + 6 – 5 = 3x + 1 = 3( -4) – 5 = -12– 5 = -17 = 3( 2) – 5 = 6 – 5 = 1 = 3( 3x) – 5 = 9x – 5 = [3(x) – 5] + [3(2) – 5] = [3x – 5] + [1] = 3x – 4

Given g: a number squared plus 6 g(x) = x ) Find g(-1)3) Find g(2a) 1) Find g(4) = ( -1) = = 7 = ( 4) = = 22 = ( 2a) = 4a ) Find g(x - 1) 4) Find 2g(a) = 2[( a) 2 + 6] = 2a = ( x-1) = x 2 – 2x = x 2 – 2x + 7

Operations on Functions Operations Notation: Sum: Difference: Product: Quotient: Example 1Add / Subtract Functions a)b)

Example 2Multiply Functions a) b)

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

Example 2Composites of a Function Set a) XY XY f(x) g(x) f(g(x)) XY This means f(g(5))=3

Example 2Composites of a Function Set b) XY f(x) XY g(x) In set form, not every x-value of a composite function is defined

Evaluate Composition Functions Find: a)f(g(3))b) g(f(-1))c) f(g(-4)) d) e) f)

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

Steps to Find Inverses [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) Inverse is not a function FunctionInverse

Example 1One-to-One (Horizontal Line Test) Determine whether the functions are one-to-one. a)b) One-to-OneNot One-to-One

Example 2Inverses of Ordered Pair Relations a) b) Are inverses f -1 (x) or g -1 (x) functions?

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

Example 3Find an Inverse Function a)b)

Example 3Continued 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 a)Method 1: Directly solve for inverse and check b)Method 2: Composition Property Yes, Inverses