Functions 3.6. Fall Math (Math 1010) M / 13

Transcription

1 Functions 3.6 Fall Math 1010 (Math 1010) M / 13

2 Roadmap Functions: Relations, Functions Evaluating Functions, Finding Domains and Ranges (Math 1010) M / 13

3 3.6 - Functions Our study of functions will come in parts: defining a function, seeing why functions stand out from all relations, using the notation of functions, and finding associated domains and ranges of functions. On 09/23 we described what parts we ve taken from graphing. Here is a sample of popular choices: points / order pairs / (x, y); intercepts; shapes - lines, curves, V -shapes; trends; axes; distance; slope between two points Many of the above ideas are useful when learning about functions. (Math 1010) M / 13

4 3.6 - Points / Ordered Pairs Thinking about points is a good place to start. Here is a relation table: x y point 0 1 (0,1) 1 3 (1,3) 2 5 (2,5) 3 5 (3,5) 0 3 (0,3) (Math 1010) M / 13

5 3.6 - Points / Ordered Pairs Thinking about points is a good place to start. Here is a relation table: x y point 0 1 (0,1) 1 3 (1,3) 2 5 (2,5) 3 5 (3,5) 0 3 (0,3) The set of x values is the domain. The set of y values is the range. Domain = {0, 1, 2, 3}, Range = {1, 3, 5} (Math 1010) M / 13

6 3.6 - Points / Ordered Pairs Thinking about points is a good place to start. Here is a relation table: x y point 0 1 (0,1) 1 3 (1,3) 2 5 (2,5) 3 5 (3,5) 0 3 (0,3) The set of x values is the domain. The set of y values is the range. Domain = {0, 1, 2, 3}, Range = {1, 3, 5} This relation cannot be a function. (Math 1010) M / 13

7 3.6 - Functions A function is a rule that pairs each element of a domain to exactly one element of the range. (Math 1010) M / 13

8 3.6 - Functions A function is a rule that pairs each element of a domain to exactly one element of the range. Note: No two ordered pairs have the same first component and different second components. That is, (2, 4) and (2, 18) cannot be ordered pairs of a function. (Math 1010) M / 13

9 3.6 - Functions A function is a rule that pairs each element of a domain to exactly one element of the range. Note: No two ordered pairs have the same first component and different second components. That is, (2, 4) and (2, 18) cannot be ordered pairs of a function. The rule is a process that turns one number into another number. For instance, the squares of all real numbers is the rule that turns 1 into 1, 1.1 into 1.21, 1.2 into 1.44, and so on. (Math 1010) M / 13

10 3.6 - Functions A function is a rule that pairs each element of a domain to exactly one element of the range. Note: No two ordered pairs have the same first component and different second components. That is, (2, 4) and (2, 18) cannot be ordered pairs of a function. The rule is a process that turns one number into another number. For instance, the squares of all real numbers is the rule that turns 1 into 1, 1.1 into 1.21, 1.2 into 1.44, and so on. It is okay to have two or more ordered pairs with different first components and the same second component. That is, (2, 6) and (40, 6) and ( 19, 6) may be ordered pairs of a function. (Math 1010) M / 13

11 3.6 - Functions A function is a rule that pairs each element of a domain to exactly one element of the range. Algebraically this is represented with an equation. For the examples below take x as the independent variable and y as the dependent variable. (Math 1010) M / 13

12 3.6 - Functions A function is a rule that pairs each element of a domain to exactly one element of the range. Algebraically this is represented with an equation. For the examples below take x as the independent variable and y as the dependent variable. 1) Is y 2 x = 0 a function? (Math 1010) M / 13

13 3.6 - Functions A function is a rule that pairs each element of a domain to exactly one element of the range. Algebraically this is represented with an equation. For the examples below take x as the independent variable and y as the dependent variable. 1) Is y 2 x = 0 a function? 2) Is x 2 y = 0 a function? (Math 1010) M / 13

14 3.6 - Functions A function is a rule that pairs each element of a domain to exactly one element of the range. Algebraically this is represented with an equation. For the examples below take x as the independent variable and y as the dependent variable. 1) Is y 2 x = 0 a function? 2) Is x 2 y = 0 a function? 3) Is 4x + 8y = 5 a function? (Math 1010) M / 13

15 3.6 - Functions A function is a rule that pairs each element of a domain to exactly one element of the range. Algebraically this is represented with an equation. For the examples below take x as the independent variable and y as the dependent variable. 1) Is y 2 x = 0 a function? 2) Is x 2 y = 0 a function? 3) Is 4x + 8y = 5 a function? 1) No, 2) Yes, 3) Yes (Math 1010) M / 13

16 3.6 - Function Notation When writing the rule of a function in equation form, it is convenient to keep track of input and output pairs and to give the function a name. This is all done with function notation. (Math 1010) M / 13

17 3.6 - Function Notation When writing the rule of a function in equation form, it is convenient to keep track of input and output pairs and to give the function a name. This is all done with function notation. Example: y = (x + 3) 3 becomes f (x) = (x + 3) 3 (Math 1010) M / 13

18 3.6 - Function Notation When writing the rule of a function in equation form, it is convenient to keep track of input and output pairs and to give the function a name. This is all done with function notation. Example: y = (x + 3) 3 becomes f (x) = (x + 3) 3 The name of the function is f. f (x) is read, f-of-x, or, the value of f at x. Evaluating a function is done by subsituting an x value into the function. Example: Evaluate the value of f (x) = (x + 3) 3 at x = 5. (Math 1010) M / 13

19 3.6 - Function Notation When writing the rule of a function in equation form, it is convenient to keep track of input and output pairs and to give the function a name. This is all done with function notation. Example: y = (x + 3) 3 becomes f (x) = (x + 3) 3 The name of the function is f. f (x) is read, f-of-x, or, the value of f at x. Evaluating a function is done by subsituting an x value into the function. Example: Evaluate the value of f (x) = (x + 3) 3 at x = 5. f ( 5) = (( 5) + 3) 3 = ( 2) 2 = 8. (Math 1010) M / 13

20 3.6 Functions Are some values of x not allowed for f (x) = 1 x 10? (Math 1010) M / 13

21 3.6 Functions Are some values of x not allowed for Are some values of x not allowed for f (x) = 1 x 10? f (x) = x? (Math 1010) M / 13

22 3.6 - Implied Domains and Ranges Implied domains are the set of real values for x that yield real values for f (x). The rule for the function implies the domain. Exclude: sets of values for x that result in division by zero. sets of values for x that result in taking the square-roots of negative numbers. Otherwise, explicit domains very directly state which x values to exclude when the rule of the function is given. Example: 7. f (x) = x 7, x 7. The domain is {x x 7} or all x except (Math 1010) M / 13

23 3.6 Implied Domain and Range Find the implied domain of f (x) = 5x 2 (Math 1010) M / 13

24 3.6 Implied Domain and Range Find the implied domain of The inequality is 5x 2 0 f (x) = 5x 2 (Math 1010) M / 13

25 3.6 Implied Domain and Range Find the implied domain of f (x) = 5x 2 The inequality is 5x 2 0 Solved: x 2 5. All real numbers x such that x 2 5 is the domain. (Math 1010) M / 13

26 3.6 Explicit Domain and Range The domain of f given below is all real numbers. What is the rule? { x, if x 0 f (x) = x, if x > 0 (Math 1010) M / 13

27 3.6 Explicit Domain and Range The domain of f given below is all real numbers. What is the rule? { x + 8, if x < 2 f (x) = 6 3x, if x 2 (Math 1010) M / 13

28 Assignment Assignment: For Wenesday: 1. Exercises from 3.6 due Wednesday, October Exam # 1: Chapter 3 & Cumulative Chapters 1-2, October 2 (Math 1010) M / 13