Section 1.6. Functions

Transcription

1 Section 1.6 Functions

2 Definitions Relation, Domain, Range, and Function The table describes a relationship between the variables x and y. This relationship is also described graphically. x y Section 1.6 Lehmann, Intermediate Algebra, 4ed Slide 2

3 Definition A relation is a set of ordered pairs. Definition The domain of a relation is the set of all values of the independent variable. Definition Definitions Relation, Domain, Range, and Function The range of the relation is the set of all values of the dependent variable. Section 1.6 Lehmann, Intermediate Algebra, 4ed Slide 3

4 Definitions Relation, Domain, Range, and Function Think of a relation as a machine where: x are the inputs -Each member of the domain is an input. y are the outputs -Each member of the range is an output. Definition A function is a relation in which each input leads to exactly one output. Section 1.6 Lehmann, Intermediate Algebra, 4ed Slide 4

5 Deciding whether an Equation Describes a Function Is the relation y = x +2 a function? Find the domain and the range of the relation. Consider some input-output pairs. Relation, Domain, Range, and Function Section 1.6 Lehmann, Intermediate Algebra, 4ed Slide 5

6 Deciding whether an Equation Describes a Function Relation, Domain, Range, and Function Continued Each input leads to just one output namely, the input increased by 2 so the relation y = x + 2 is a function. Domain We can add 2 to any real number. So, the domain is the set off all real numbers. Range Output is two more than the input. So, the range is the set off all real numbers. Section 1.6 Lehmann, Intermediate Algebra, 4ed Slide 6

7 Deciding whether an Equation Describes a Function Relation, Domain, Range, and Function Is the relation y = ± xa function? If x = 1, then y = ±1 Input x = 1 leads to two outputs: y = 1 and y = 1 Therefore, the relation y = ± x is not a function Section 1.6 Lehmann, Intermediate Algebra, 4ed Slide 7

8 Deciding whether an Equation Describes a Function Is the table a function? Section 1.6 Relation, Domain, Range, and Function Consider the input x = 4 Substitute 4 for x and solve for y: Input x = 4 leads to two outputs: y = 2 and y = 2 2 So, the relation y = xis not a function ± Lehmann, Intermediate Algebra, 4ed Slide 8

9 Deciding whether a Table is a Function Relation, Domain, Range, and Function Is the relation 2 y = Input x = 1 leads to two outputs y = 3 and y = 5 So, the relation is not a function. x a function? x y Section 1.6 Lehmann, Intermediate Algebra, 4ed Slide 9

10 Is the relation described by the graph a function? Definition Relation, Domain, Range, and Function The input x = 1 leads to two outputs: y = 4 and y = 4 So, the relation is not a function Section 1.6 Lehmann, Intermediate Algebra, 4ed Slide 10

11 Deciding whether a Graph Describes a Function The vertical line sketched intersects the circle more than once The relation is not a function. Vertical Line Test Is the relation described by the graph on in the next slide a function? Section 1.6 Lehmann, Intermediate Algebra, 4ed Slide 11

12 Deciding whether a Graph Describes a Function Vertical Line Test All vertical lines intersects the curve at one point Slide 12

13 Deciding whether an Equation Describes a Function Sketch the graph of y = 2x+ 1 Vertical Line Test Is the relation y = 2x + 1 a function? Each vertical line would intersect at just one point So, the relation is a function Slide 13

14 Definition Definition and Properties Linear Function A linear function is a relation whose equation can be put into the form y = mx + b where m and b are constants. Properties Properties of linear functions: 1. The graph of the function is a nonvertical line. Slide 14

15 Properties of Linear Functions Linear Function 2. The constant m is the slope of the line, a measure of the line s steepness. 3. If m > 0, the graph of the function is an increasing line. 4. If m < 0, the graph of the function is a decreasing line. 5. If m = 0, the graph of the function is a horizontal line. Slide 15

16 Properties of Linear Functions Linear Function 6. If an input increases by 1, then the corresponding output changes by the slope m. 7. If the run is 1, the rise is the slope m. 8. The y-intercept of the line is (0, b). Since a linear equation of the form y = mx + b is a function, we know that each input leads to exactly out output. Slide 16

17 Definition Definition Rule of Four for Functions We can describe some or all of the input output pairs of a function by means of 1. an equation 2. a graph 3. a table, or 4. words. These four ways to describe input output pairs of a function are known as the Rule of Four for functions. Slide 17

18 Describing a Function by Using the Rule of Four Is the relation Rule of Four for Functions y = 2x 1 a function? Since y = 2x 1is of the form y = mx + b, it is a (linear) function. List some input output pairs of a table. y = 2x 1by using Slide 18

19 Describing a Function by Using the Rule of Four We list five input output pairs of y = 2x 1in the table on the right. Rule of Four for Functions Describe the input output pairs of y = 2x 1using a graph. x y 2 2( 2) 1 = 3 1 2( 1) 1 = 1 0 2(0) 1 = 1 1 2(1) 1 = 3 2 2(2) 1 = 5 Slide 19

20 Describing a Function by Using the Rule of Four We graph the right. y Rule of Four for Functions = 2x 1 on Describe the input output pairs of y = 2x 1by using words. For each input output pair, the output is 1 less than 2 times the input. Slide 20

21 Finding the Domain and Range Using a Graph to Find the Domain and Range of a Function Using the graph of the function to determine the function s domain and range. Domain is the set of all x coordinates of the graph No breaks in the graph Leftmost point: ( 4, 2),the rightmost point :(5, 3), Domain is 4 x 5. Slide 21

22 Finding the Domain and Range Using a Graph to Find the Domain and Range of a Function Continued Range is the set of all y-coordinates of points Lowest point is (5, 3), highest is (2, 4) The range is 3 y 4 Using the graph of the function to determine the function s domain and range. Slide 22

23 Finding the Domain and Range Using a Graph to Find the Domain and Range of a Function Domain Extends left and right indefinitely without breaks Domain: set of all real numbers Range Lowest point is (1, 3) Highest is indefinite without breaks Range: y 3 Slide 23