Deep and Wide: Domain & Range The student is expected to identify mathematical domains and ranges and determine reasonable domain and range values for given situations.
A function is a set of ordered pairs of numbers (x, y) such that no x –values are repeated. What are the domain and range of a function? The domain and range of a function are sets that describe those ordered pairs.
Definition Example {(3, 6), (2, 8), (5, 3)} { 3, 2, 5} { 6, 8, 3} Domain All the x-coordinates in the function's ordered pairs { 3, 2, 5} Range All the y-coordinates in the function's ordered pairs { 6, 8, 3}
The domain is the set of all the values of the independent variable, the x-coordinate The range is the set of all the values of the dependent variable, the y-coordinate.
Identify the domain and range of the function below. { 2, 7), (4, 11), (6, 15), (8, 19)} The domain is { 2, 4, 6, 8} The range is { 7, 11, 15, 19}
Graphs The domain of a function is the set of all the x-coordinates in the functions’ graph Domain 3 ≤ x ≤ 12
The range of a function is the set of all the y-coordinates in the functions’ graph Range 6 ≤ y ≤ 12
What is the domain of this function? What is the range of this function? Domain is 0 ≤ x ≤ 4 Range is 1 ≤ y ≤ 5 Answer: Domain is 0 ≤ x ≤ 4 Range is 1 ≤ y ≤ 5
The graph shows the path of a golf ball What is the range of this function? F 0 < y < 100 G 0 ≤ y ≤ 100 H 0 ≤ x ≤ 5 J 0 ≤ x ≤ 5 Answer is G Is this function continuous or discrete? Would it be reasonable to have a range with negative numbers?
What is the domain of this function? A -1 ≤ x ≤ 5 B -1 ≤ x ≤ 9 C 2 ≤ x ≤ 5 D 0 ≤ y ≤ 9 Students should immediately realize that “D” cannot be the answer because y is the dependent variable and therefore y represents the RANGE, not the DOMAIN. Answer is A
What is the domain of the function shown on the graph? A -2 < y ≤ 2 B -4 ≤ x ≤ 6 C -4 < y ≤ 2 D -2 < x ≤ 6 Students should immediately realize that A and C cannot be the answer. We are looking for domain. Therefore, our variable will be an X, NOT A Y.
Sometimes you will be asked to determine a REASONABLE domain or range
The average daily high temperature for the month of May is represented by the function t = 0.2n + 80 Where n is the date of the month. May has 31 days. What is a reasonable estimate of the domain? Answer: 1 ≤ n ≤ 31 What is a reasonable estimate of the range Answer: See next slide
Our function rule is: t = 0.2n + 80 Our domain is 1 ≤ n ≤ 31 Our smallest possible n is 1 Our largest possible n is 31 To find the range, substitute 1 into the equation and solve. Then substitute 31 into the equation and solve.
Our function rule is: t = 0.2n + 80 Substitute a 1 t = 0.2n + 80 t = 0.2(1) + 80 t = 0.2 + 80 t = 80.2 Substitute a 31 t = 0.2n + 80 t = 0.2(31) + 80 t = 6.2 + 80 t = 86.2 Reasonable range is 80.2 ≤ t ≤ 86.2