A function from a set A to a set B is a relation that assigns to each element x in the set A exactly one element y in the set B. The set A is called the domain of the function. The set B is called the range of the function. X Y 1 5 2 10 3 15 4 B 4 2 A 1 6 8 1 7 3 y – 3x = 10

Example: Determine Functions Example: Determine whether the relation represents y as a function of x. a) {(-2, 3), (0, 0), (2, 3), (4, -1)} Function b) {(-1, 1), (-1, -1), (0, 3), (2, 4)} Not a Function y depends on value of x, y is dependent variable x is the independent variable f(x) is dependent variable y and f(x) are the same Example: Determine Functions

Tests for function vertical line test no vertical line intersects graph at more than one point If there is a y variable with largest exponents being odd, usually it is a function (even exponents usually are not) All first members of ordered pairs are different

Is y a function of x? y is not a function of x, is 3x + y = 5? Solve for y y = 5 - 3x (yes, y is a function of x) y is not a function of x, y2 = 25 - x2 if x = 3 then y = 4 or y = -4 y is not a function of x (even exponent)

To determine if equation defines a function, f(x) notation f(x) = 3x - 1 used to name a function f is name of function x is independent variable 3x - 1 is rule used to evaluate y = f(x) To determine if equation defines a function, 1. solve equation for dependent variable (y) 2. determine if each single value of independent variable (x) produces exactly one value of dependent (y) variable * normally x is independent variable while y is dependent

Example: Let f (x) = x2 – 3x – 1. Find f (–2). To evaluate a function f (x) at x = a, substitute the specified value a for x into the given function. Example: Let f (x) = x2 – 3x – 1. Find f (–2). f (–2) = (–2)2 – 3(–2) – 1 f (–2) = 9 Example: Let f (x) = 4x – x2. Find f (x + 2). f (x + 2) = 4(x + 2) – (x + 2)2 f (x + 2) = 4x + 8 – (x2 + 4x + 4) f (x + 2) = 4x + 8 – x2 – 4x – 4 f (x + 2) = 4 – x2 Evaluating Functions

In a piecewise-defined function, you are given two or more functions to work with followed by defined domains for each function. You need to decide which domain the value you are using fall in and use your value with that function. Ex. Evaluate the function when x = 5 and -3 When x = 5, we us the bottom function 5 – 1 = 4 so f(5) = 4 When x = –3, we us the top function (-3)2 + 1 = 10 so f(-3) = 10

Example: Find the domain of the function f (x) = 3x +5 The domain of a function f is the set of all real numbers for which the function makes sense. Example: Find the domain of the function f (x) = 3x +5 Domain: All real numbers Example: Find the domain of the function The function is defined only for x-values for which x – 3  0. Solving the inequality yields x – 3  0 x  3 Domain: {x| x  3} Definition of Domain

Example: Find the domain of the function The x values for which the function is undefined are excluded from the domain. The function is undefined when x2 – 1 = 0. x2 – 1 = 0 (x + 1)(x – 1) = 0 x =  1 Domain: {x| x   1} Example: Find Domain

Restrictions on domain Domain usually all real numbers f(x) = (3 - x) 3-x > 0 so x < 3 written {x|x < 3} = D Another exclusion of numbers g(x) = 4/ (x-2), x  2 so {x|x  2}=D