Functions MATHPOWERTM 11, WESTERN EDITION 5.5.1

A function is a relation in which no two ordered pairs have Functions A function is a relation in which no two ordered pairs have the same first coordinate. For every x there is only one y. (1, 2) (2, 4) (3, 6) (4, 8) A relation that is a FUNCTION (1, 2) (2, 4) (2, 5) (3, 6) A RELATION that is not a function 1 2 3 2 3 4 1 2 3 2 3 4 1 2 3 2 3 4 FUNCTION FUNCTION RELATION 5.5.2

Vertical Line Test: If no two points on a graph can Functions Vertical Line Test: If no two points on a graph can be joined by a vertical line, the graph is a function. Function Relation Function 5.5.3

An equation that is a function may be expressed Functional Notation An equation that is a function may be expressed using functional notation. The notation f(x) (read “f at (x)”) represents the variable y. E.g., y = 2x + 6 can be written as f(x) = 2x + 6. Given the equation y = 2x + 6, evaluate when x = 3. y = 2(3) + 6 y = 12 For the function f(x) = 2x + 6, the notation f(3) means that the variable x is replaced with the value of 3. f(x) = 2x + 6 f(3) = 2(3) + 6 f(3) = 12 5.5.4

Given f(x) = 4x + 8, find each: Evaluating a Function Given f(x) = 4x + 8, find each: 1. f(2) 2. f(a) f(a) = 4(a) + 8 = 4a + 8 f(2) = 4(2) + 8 = 16 3. f(a + 1) 4. f(-4a) f(a + 1) = 4(a + 1) + 8 = 4a + 4 + 8 = 4a + 12 f(-4a) = 4(-4a) + 8 = -16a+ 8 5.5.5

If f(x) = 3x - 1 and g(x) = 5x + 3, find each: Evaluating a Function If f(x) = 3x - 1 and g(x) = 5x + 3, find each: 1. f(2) + g(3) 2. f(4) - g(-2) = [3(2) -1] + [5(3) + 3] = 6 - 1 + 15 + 3 = 23 = [3(4) - 1] - [5(-2) + 3] = 11 - (-7) = 18 3. 3f(1) + 2g(2) = 3[3(1) - 1] + 2[5(2) + 3] = 6 + 26 = 32 5.5.6

If g(x) = 2x2 + x - 3, find each: Evaluating a Function If g(x) = 2x2 + x - 3, find each: 1. g(2) g(2) = 2(2)2 + 2 - 3 = 8 + 2 - 3 = 7 2. g(x + 1) g(x + 1) = 2(x + 1)2 + (x + 1) - 3 = 2(x2 + 2x + 1) + x + 1 - 3 = 2x2 + 4x + 2 + x - 2 = 2x2 + 5x 5.5.7