6.1 Relations and Functions Math 8CP

A relation is a set of ordered pairs. A relation can be represented by: a graph list of ordered pairs a table

Ordered Pairs Graph (4, 3) (-2, -1) (-3, 2) (2, -4) (0, -4) Table x y (4, 3) (-2, -1) (-3, 2) (2, -4) (0, -4) Table x y 4 3 -2 -1 -3 2 -4

Domain: {-3, -1, 2, 4} Range: { -4, -3, 0, 1} Ex. A: Express the relation { (-1, 0), (2, -4), (-3, 1), (4, -3)} as a table, and a graph. Then determine the domain and range. x y x y -3 1 -1 2 -4 4 Domain: all inputs; values of x Range: all outputs; values of y Domain: {-3, -1, 2, 4} Range: { -4, -3, 0, 1}

Function: A relation between two quantities, called the input and the output. For each input, there is exactly one output. Multiple inputs may give the same output. Huh?

Let’s say you go to Baja Fresh. With your lunch you order a medium Coke. Will you see two different prices for a medium Coke on the menu? No. That is what’s meant by “for each input” (medium Coke), “there is exactly one output” (one price for the medium Coke). You notice also that other medium sodas like Sprite also cost the same as a medium Coke. That is what’s meant by “even though different inputs” (medium Coke, medium Sprite) “may give the same output” (same price for both items).

Is this relation a function? One way to look at functions is to make an input/output table such as the one below. Is this relation a function? x y 1 2 -4 3 4 -3 5 10 6 12 This relation is a function. For every x-value (input) there is exactly one y-value (output).

Is this relation a function? x y 1 2 4 8 10 6 12 No, it is not a function. Why? Because the input 2 had two different outputs. Think of it like this: 2 Cokes can’t be $4 and $10.

Ex. B: Does the table represent a function. Explain Ex. B: Does the table represent a function? Explain. If it is a function, name the domain and the range. x y 1 4 2 6 3 8 10 5 12 14 The table DOES represent a function because for every input there is only one output. Domain: {1, 2, 3, 4, 5, 6} Range: {4, 6, 8, 10, 12, 14}

Ex. C: Does the table represent a function. Explain Ex. C: Does the table represent a function? Explain. If it is a function, name the domain and the range. x y -3 4 -2 6 -1 8 The table DOES represent a function because for every input there is only one output. Domain: {-3, -2, -1} Range: {4, 6, 8}

Ex. D: Does the table represent a function? Explain. y 5 3 7 2 9 4 No, it is not a function. Why? Because the input 5 had two different outputs. Think of it like this: 5 Cokes can’t be $3 and $4.

Ex. E: Does the table represent a function. Explain Ex. E: Does the table represent a function? Explain. If it is a function, name the domain and the range. x y -3 8 -2 6 -1 4 2 1 The table DOES represent a function because for every input there is only one output. Domain: {-3, -2, -1, 0, 1, 2} Range: {2, 4, 6, 8}

A vertical line test is used to determine whether a graph represents a function. x (input) y (output) A graph is a function if any vertical line intersects the graph at no more than one point. The graph is a function.

You can use your pencil to check if a graph is a function You can use your pencil to check if a graph is a function. Keep your pencil straight to represent a vertical line and pass it across the graph. If it touches the graph at more than one point, the graph is not a function. x (input) y (output)

x (input) y (output) No, it is not a function.

x (input) y (output) No, it is not a function.

x (input) y (output) Yes, it is a function.

x (input) y (output) No, it is not a function.

x (input) y (output) Yes, it is a function.

. . . . . Ex. F: Make a table and a graph for: y = 2x + 1 Is it a function? Explain. . . x y -2 -1 1 2 . -3 . -1 . x (input) 1 3 5 y (output) Yes, the relation is a function. For each input there is one output, and it passes the Vertical Line Test.