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Notes: Relations and Functions Learning Targets: 2.1- I can describe ordered pairs using the coordinate system, a table, and mapping. 2.2- I can analyze the domain and range of relations and functions. 2.3- I can determine if a relation is a function. 2.4- I can explain the parts of a Cartesian plane in respect to relations and functions. (Throughout lesson) 2.5- I can create multiple representations of relations and functions. 2.6- I can evaluate functions. Coordinate Plane: grid formed by the intersection of two number lines, the horizontal axis (x-axis) and the vertical axis (y-axis). Ordered Pair: set of numbers, or coordinates, written in the form (x, y). X-Coordinate/X-Value/Input/Domain: first number in each ordered pair. Y-Coordinate/Y-Value/Output/Range: second number in each ordered pair. Independent Variable: (Domain/x-coordinates) the variable that determines the output. Dependent Variable: (Range/y-coordinates) the variable with a value that depend on the value of the independent variable. (Remember: y depends on x) Coordinate Plane/Cartesian Plane: Label each quadrant, the x and y axis, and the origin. Write the ordered pair for each point shown in the coordinate plane below (point a thru point k). Then name the quadrant in which each point is located. Relation: a set of ordered pairs. A Relation can be represented in several ways: o o o o Representations of Relations Ordered Pair: Table: (-1, 4) (0, 2) (2, 5) (5, 1) Graph: Mapping: Illustrates how each element of the Domain is paired with an element in the range. Practice: Create a table, graph, and mapping for the following relation below. o { (2, -5), (3, 2), (-1, 4), (0, -1) } Table: Graph: Mapping: Domain: ________________________________________________________________________ Range: _________________________________________________________________________ Example: {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3)} Domain: {1, 2} Range: {1, 2, 3} State the domain and range of the relations below. 1. {(2, 1), (4, 2), (3, 3), (4, 1)} 2. {(1, 4), (2, -3), (-2, 1), (0, 3)} 3. {(2, 0), (-1, 3), (2, 1), (1, 3)} D= D= D= R= R= R= Function: A relation in which each element of the domain is paired with exactly one element of the range. Focus on the x-coordinates, when given a relation If the set of ordered pairs have different x-coordinates, it IS A function. If the set of ordered pairs have same x-coordinates, it is NOT a function. Y-coordinates have no bearing in determining functions ***If the relation is a function, then the domain doesn’t share. A function can be represented as a: o o o o o Representations of Functions -Create a table, set of ordered pairs, mapping, and a graph for the equation below. Equation: 𝑦 = 2𝑥 + 2 *To graph any function (equation)1. 2. 3. Make a table of values (X/Y table) made up of chosen X-values. Substitute your X-values into the equation to find the corresponding Y-values. Graph Table: Ordered Pairs: Mapping: Graph: Identifying Functions: Determine whether the following is a function. Ex.1) {(0, -5), (1, -4), (2, -3), (3, -2), (4, -1), (5, 0)} Ex.2) {(-1, -7), (1, 0), (2, -3), (0, -8), (0, 5), (-2, -1)} Ex.3) Ex.4) Ex.5) Ex.6) X -5 -2 4 2 X -2 3 1 3 Y 2 -1 2 0 Y -5 3 4 1 Vertical Line Test- a relation is a function if a vertical line intersects the graph only once. If the vertical line intersects the graph more than once, the graph is NOT a function. AKA: “The Pencil Test”- Take a pencil and move it from left to right (–x to x); if it crosses more than one point, it is not a function Use the vertical line test to determine if the following graphs represent a function. Example 1: Example 2: Example 3: Real World Example: Which situation represents a function? a) The items in a store to their prices on a certain date. b) Types of fruits to their colors Extra Practice: Determine if relations below are functions? 1. Names and social security numbers. 2. Addresses and names. Discrete Function- A graph that consists of points that are not connected. Continuous Function- A function graphed with a line of smooth curve. Determine if the following functions are discrete or continuous. Then identify the domain and range. Ex. 1) Ex. 2) Type of Graph: ______________ Type of Graph: _______________ Domain: ___________________ Domain: ____________________ Range: ____________________ Range: _____________________ Ex. 3) Ex. 4) Type of Graph: ______________ Type of Graph: _______________ Domain: ___________________ Domain: ____________________ Range: ____________________ Range: _____________________ EVALUATING FUNCTIONS: Function Notation- Equations that are functions can be written in function notation. Equation Function Notation Y = 3x - 8 f(x) = 3x - 8 𝒇(𝒙) means function of 𝑥 and is read “𝒇of 𝒙.” In a function, 𝒙 represents the elements of the domain, and 𝒇(𝒙) represents the elements of the range. Ex.) Suppose you want to find the value in the range that corresponds to the element 1 in the domain. EX) For 𝑓(𝑥) = 2𝑥 + 1, find the value of 𝑓(1). The notation f(1) means to replace x with 1. 𝑓(𝑥) = 2𝑥 + 1 𝑓(1) = Ex.) Given 𝑔(𝑥) = 𝑥 2 − 3, find 𝑔(−2). Given 𝑓(𝑥) = 2𝑥 2 − 3𝑥, find the following. a) 𝑓(3) Extra Practice: Evaluate the following. 1. For 𝑓(𝑥) = −4𝑥 + 7, find 𝑓(2). b) 3(𝑓(𝑥)) 2. For 𝑔(𝑥) = 2𝑥 2 − 5, find 𝑔(−3). 3. For 𝑓(𝑥) = 3𝑥 + 2, find 𝑓(4) − 5. 4. For 𝑔(𝑥) = −2𝑥 + 4, find 𝑔(ℎ + 5) Real World Connection: The table below shows the masses, m grams, of different numbers of identical marbles, n. Number of Marbles (n) Mass of Marbles (m grams) 1 1.27 2 2.54 3 3.81 4 5.08 5 6.35 6 7.62 A. B. C. D. Why is the relation also a function? Identify the independent variable and the dependent variable. Justify your choices. Create a set of ordered pairs, a mapping, a graph, and an equation for the function above. Using the equation you created in part C, find the mass of 12 marbles, or 𝑓(12). A. _________________________________________________________________________________________ ___________________________________________________________________________________________ B. __________________________________________________________________________________________ ____________________________________________________________________________________________ C. Ordered Pairs D. 𝑓(𝑥) = 𝑓(12) = Mapping Graph Equation