2-1: Graphing Linear Relations and Functions Objectives: Understand, draw, and determine if a relation is a function. Graph & write linear equations, determine domain and range. Understand function notation NOTES CH 2.1 Functions

Relations & Functions- I. VOCAB Relation: a set of ordered pairs Domain: the set of x-coordinates (all the values that x can be!) Range: the set of y-coordinates (all the values that y can be!) **When writing the domain and range, do not repeat values.** NOTES CH 2.1 Functions

Relations and Functions EX 1: Given the relation: {(2, -6), (1, 4), (2, 4), (0,0), (1, -6), (3, 0)} State the domain: D: {0,1, 2, 3} State the range: R: {-6, 0, 4} NOTES CH 2.1 Functions

Relations and Functions II. Representations Relations can be written in several ways: ordered pairs, table, graph, or mapping. We have already seen relations represented as ordered pairs. NOTES CH 2.1 Functions

Table EX 2: {(3, 4), (7, 2), (0, -1), (-2, 2), (-5, 0), (3, 3)} NOTES CH 2.1 Functions

Mapping Create two ovals with the domain on the left and the range on the right. Elements are not repeated. Connect elements of the domain with the corresponding elements in the range by drawing an arrow. NOTES CH 2.1 Functions

Mapping EX 3: {(2, -6), (1, 4), (2, 4), (0, 0), (1, -6), (3, 0)} D R 2  -6 4 2 1 3 NOTES CH 2.1 Functions

Mapping diagrams Make an mapping diagram. EX 4: {(1, 2), (-1, 4), (0, 2), (4, -7)} See board. EX 5: {(2, 3), (-1, -5), (2, 6)} See board. NOTES CH 2.1 Functions

III. Functions A function is a relation in which the members of the domain (x- values) DO NOT repeat. So, for every x-value there is only one y-value that corresponds to it. y-values can be repeated. NOTES CH 2.1 Functions

Do the ordered pairs represent a function? Make a mapping diagram. EX 6. {(3, 4), (7, 2), (0, -1), (-2, 2), (-5, 0), (3, 3)} No, 3 is repeated in the domain. EX 7. {(4, 1), (5, 2), (8, 2), (9, 8)} Yes, no x-coordinate is repeated. NOTES CH 2.1 Functions

When given a graph, is it a function? Vertical Line Test: If a vertical line is passed over the graph and it intersects the graph in exactly one point, the graph represents a function. Domain: all values x can be. “Where does the graph stop to the left and to the right?” Range: all values y can be. “ Where does the graph stop looking up and down?” NOTES CH 2.1 Functions

Does the graph represent a function? Name the domain and range. x y Yes D: all reals R: all reals R: y ≥ -6 EX 8 x y EX 9 NOTES CH 2.1 Functions

Does the graph represent a function? Name the domain and range. x y No D: x ≥ 1/2 R: all reals D: all reals EX 10 x y EX11 NOTES CH 2.1 Functions

Does the graph represent a function? Name the domain and range. x y Yes D: all reals R: y ≥ -6 No D: x = 2 R: all reals EX12 x y EX 13 NOTES CH 2.1 Functions

IV. Function Notation When we know that a relation is a function, the “y” in the equation can be replaced with f(x). f(x) is simply a notation to designate a function. It is pronounced ‘f’ of ‘x’. The ‘f’ names the function, the ‘x’ tells the variable that is being used. NOTES CH 2.1 Functions

Value of a Function Since the equation y = x - 2 represents a function, we can also write it as f(x) = x - 2. EX 14. Find f(4): f(4) = 4 - 2 f(4) = 2 NOTES CH 2.1 Functions

Value of a Function EX 15. If g(s) = 2s + 3, find g(-2). =-4 + 3 = -1 g(-2) = -1 NOTES CH 2.1 Functions

Value of a Function EX 16. If h(x) = x2 - x + 7, find h(2c). h(2c) = (2c)2 – (2c) + 7 = 4c2 - 2c + 7 NOTES CH 2.1 Functions