Relations and Functions April S. Brown MAT 250. Introduction Brief history behind relations and functions Goal 3.02 – Identifying the properties and relationships.

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Presentation transcript:

Relations and Functions April S. Brown MAT 250

Introduction Brief history behind relations and functions Goal 3.02 – Identifying the properties and relationships of data in tables, graphs and equations Goal Exploring the differences between relations and functions

Brief History 1694: The word function was first used by GW Liebniz. 1698: J. Bernoulli defined a function as “any expression involving variables and constants.” 1734: The familiar notation f(x) was first used by Euler. ONLY RECENTLY has it become standard practice to treat a function as a relation with special properties.

Relations Relation – a set of ordered pairs (x, y) Idea of “is related to” between x and y Examples: (square, 4)shape and number of sides (April, 30)month and number of days

Functions Function of x – a relation in which no two ordered pairs have the same x-value Examples: (5, 5) and (5, 2)not a function (3, 5) and (5, 2)function (13, -24) and (13, 76)not a function (1, 24) and (7, 24)function

The Vertical Line Test When we have a graph of coordinates, we can perform a test on it to determine if it is a function or a relation Using a straight edge, look at the function from left to right – does the straight edge touch the graph in more than one place?

Relations The vertical line test fails – this is a relation

Functions The vertical line test does not fail in any place – this is a function

Proof of Functions See notes – 2/25/02 By contradiction By contraposition

Function Notation When a function is defined by an equation, it is convenient to name the function The set of ordered pairs (x, y) that satisfy the equation form a function Example: y = 3x + 2 (x, y) is a “solution”

Domain and Range Domain – the x-values of a function Range – the y-values of a function Examples: function coordinates (17, 2), (24, 18), (2, 3), (5, 9) domain = 17, 24, 2, 5 range = 2, 18, 3, 9

Image and Pre-Image Another way to look at functions is by the image and pre-image definitions Function notation: f(x) = y y is “the value of f at x” y is “the image of x under f” x is “the pre-image of y under f”

More In-Depth Look The domain is the value that “goes into” a function The range is the value that “comes out” of a function The Function Machine Website the function machine

Independent / Guided Practice Work in small groups Categories of Problems: - using tables, graphs and data sets, identify functions and relations - give some examples of relations in the real world - use the vertical line test - given a set of coordinates, determine the domain and range, and “plug them into” the equations given

Summary Review definitions - Relation - Function - Vertical line test - Function notation - Domain - Range - Image - Pre-image

Closing Today’s topics: - The history behind relations and functions - Properties and relationships of data in tables, graphs and equations - Exploring the differences between relations and functions - Practice to reinforce concepts