Functions and Their Properties Section 1.2 Day 1

Functions A real-valued function f is a rule that assigns to each real number x in a set X of numbers, a unique real number y in a second set Y of numbers. The set X of all input values is called the domain of the function f and the second set Y of all output values is called the range of f.

To indicate that y comes from the function acting on x, we use function notation y = f (x). f (x) is read “f of x.” The x is the independent variable and y is the dependent variable. Functions

DOMAINRANGE X Y f x2x2 x1x1 x3x3 y2y2 y1y1 A way to picture a function is by an arrow diagram Function

DOMAINRANGE X Y f x2x2 x1x1 x3x3 y2y2 y1y1 A way to picture a function is by an arrow diagram y3y3 NOT A FUNCTION

Defining a Function Does the formula y = x 2 define y as a function of x? Yes, y is a function of x because we can rewrite it as y = f(x) so f(x) = x 2

Algebraically Defined Function is a function. Example: Is a function represented by a formula? It has the format y = f (x) = “expression in x” Substitute 5 for x

Graph of a Function Vertical Line Test: The graph of a function can be crossed at most once by any vertical line. FunctionNot a Function It is crossed more than once.

x y

x y

The domain of a function is not always specified explicitly. Unless we are dealing with a model (like volume) that necessitates a restricted domain, we will assume that the domain of a function defined by an algebraic expression is the same as the domain of the algebraic expression, the implied domain. For models, we will use a domain that fits the situation, the relevant domain. Note on Domains

Find the domain of the following functions: A) B) Domain is all real numbers but (-∞, 3) U (3, ∞) (-∞, ∞)

C) Square root is real only for nonnegative numbers.

Domain Find the domain of each of these functions a. b. c.

Support Graphically

Range Find the range of the function

Continuity We can introduce another characteristic of functions  that of continuity. We can understand continuity in several ways: (1) a continuous process is one that takes place gradually, smoothly, without interruptions or abrupt changes (2) a function is continuous if you can take your pencil and can trace over the graph with one uninterrupted motion Continuous at x = a if Discontinuous at x = a if it is not continuous at x = a

Types of Discontinuities (I) Jump Discontinuities: ex We notice our function values "jump" from 4 to 0

Types of Discontinuities (II) Infinite Discontinuities ex.

Types of Discontinuities (III) Removable Discontinuities Ex “Hole” in the graph

Continuity

Describe the increasing and decreasing behavior. The function is decreasing over the entire real line. Increasing and Decreasing Functions

Describe the increasing and decreasing behavior. The function is decreasing on the interval increasing on the interval decreasing on the interval increasing on the interval

Increasing and Decreasing Functions Describe the increasing and decreasing behavior. The function is increasing on the interval constant on the interval decreasing on the interval

Boundedness

BOUNDEDNESS

Example Identify each of these functions as bounded below, bounded above, or bounded. 1. f(x) = 3x 2 – 4 2.