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: Math 3 Hon: Unit 2

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

Examples of a Function #1: Graphs #4: Mapping #2: Table #3: Set -2 1 - 6 8 - 4 2 x y -1 -8 3 4 7 6 13 16 { (2,3), (4,6), (7,8), (-1,2), (0,4), (-2, 5), (-3, -2)} Math 3 Hon: Unit 2

Non – Examples of a Function #1: Graphs #2: Table #4: Mapping x -1 2 1 y -5 3 4 4 -2 1 8 -4 2 -1 #3: Set {(-1,2), (1,3), (-3, -1), (1, 4), (-4, -2), (2, 0)}

Practice: Is it a Function? {(2,3), (-2,4), (3,5), (-1,-1), (2, -5)} {(1,4), (-1,3), (5, 3), (-2,4), (3, 5)} 3. 5. 4. 6. -3 4 2 1 -5 9 -2 x 3 2 5 -1 y -2 4 8 Math 3 Hon: Unit 2

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 Examples: g(x) = 3x2 – 7x Reads as “g of x is 3 times x squared minus 7 times x ” g(-1) = 3(-1)2 – 7(-1) = 10 Math 3 Hon: Unit 2 6

Given f: a number multiplied by 3 minus 5 f(x) = 3x – 5 1) Find f(-4) 2) Find f(2) 3) Find f(3x) 4) Find f(x + 2) 5) Find f(x) + f(2)

Given g: a number squared plus 6 g(x) = x2 + 6 1) Find g(4) 2) Find g(-1) 3) Find g(2a) 4) Find 2g(a) 5) Find g(x - 1) Math 3 Hon: Unit 2 8

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

Example 2 Multiply Functions b)

Combining a function within another function. Notation: 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

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

Evaluate Composition Functions Find: 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 Function  Inverse Function  Inverse Inverse Notation: Inverse Properties: 1] 2]

Steps to Find Inverses One-to-One: [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 One-to-One (Horizontal Line Test) Determine whether the functions are one-to-one. a) b) Example 2 Inverses of Ordered Pair Relations Are inverses f-1(x) or g-1(x) functions? a) b)

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

Example 3 Find an Inverse Function b) c) d)

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