Section 5 2 Relations and Functions SPI 23

Section 5 -2 Relations and Functions SPI 23 C: determine the domain/range of a function represented by the graph of real-world situations Objectives: • Identify relations and functions • Evaluate Functions Vocabulary Domain: • Set of first coordinates in an ordered pair. (the x values) Range: • Set of second coordinates in a ordered pair (the y values) Relation: • Set of all ordered pairs; x value may be repeated Function f(x) or y: • a relation that assigns exactly one value in the range to each value in the domain. (x value may not be repeated)

Domain and Range Giraffes Write values from table as an ordered pair Age Domain (years Height Range (meters) 18 4. 25 (18, 4. 25) 20 4. 40 21 5. 25 14 5. 00 18 4. 85 (20, (21, (14, (18, 4. 40) 5. 25) 5. 00) 4. 85) Domain • x-values in a table or ordered pair • write values only once in order from least to greatest D={14, 18, 20, 21} Domain Range Age Height Range • y-values in a table or ordered pair • write values only once in order from least to greatest R={4. 25, 4. 40, 4. 85, 5. 00, 5. 25}

Relation and Functions Giraffes Age (years Height (meters) 18 4. 25 20 4. 40 21 5. 25 14 5. 00 18 4. 85 Relation • Set of all ordered pairs • x-value may be repeated • The (age, height) ordered pair form a relation. Function • A special relation • x-value may not be repeated • The (age, height) ordered pair does NOT form a relation.

Determine if a Relation is a Function (Vertical Line Test) Vertical Line Test Use to determine if a relation is a function Plot the data from a table of values, on a coordinate plane. x-values y-values (Domain) (Range) 3 0 -2 1 0 -1 -3 -2 3 2 Pass a pencil, vertically, across the graph. If it passes through only one point at a time, then it is a function.

Vertical Line Test Example: Use the vertical line test to determine if the graph represents a function.

Vertical Line Test Example: Does the graph represent a function? Yes, because it passes the vertical line test since the vertical line touches only 1 point at a time.

Vertical Line Test Example 2: Use the vertical line test to determine if the graph represents a function.

Vertical Line Test Example 2: Does the graph represent a function? No, because it fails the vertical line test since the vertical line touches more than 1 point at a time.

Determine if a Relation is a Function (Mapping Diagram) Mapping Diagram Also used to determine if a relation is a function List domain and range values in order x-values y-values (Domain) (Range) 3 0 -2 1 0 -1 -3 -2 3 2 Domain Range -3 -2 0 3 -2 -1 0 1 2 Draw arrow from domain values to corresponding range values Notice the domain value 3 corresponds to 2 range values

Determine if a Relation is a Function (Mapping Diagram) Which mapping diagram is a function? Domain Range 11 12 13 20 -2 -1 7 Domain Range -2 -1 4 -2 1 3

Investigate Functions Domain Range (x value) (y value) INPUT (domain, x value) x=0 Function rule x=1 f(1) f(0) == 2(1) 2(0) ++ 44 f(x)==64 2 x + 4 f(1) f(0)= y = 2 x + 4 0 4 1 6 Output (range, y value) Each x value has one and only one y value.

A Little Practice 1. Evaluate the function rules: 1. f(n) = -3 n – 10 for n = 6 f(6) = -3(6) – 10 = -18 – 10 = -28 2. y = -2 x 2 + 7 for x = - 4 y = -3(-4)2 + 7 = -3(16) + 7 = -41 2. What is the range of the function in problem 1 for the domain {0, 1, 2}? f(0) = -3(0) – 10 = 0 – 10 = -10 f(1) = -3(1) – 10 = -3 – 10 = -13 f(2) = -3(2) – 10 = -6 – 10 = -16 The range is {-16, -13, -10}.