1 / 23

Understanding Relations, Domains, Ranges, and Functions

Learn about relations, domains, ranges, and functions, and how to determine if a relation is a function using equations, graphs, and tables.

bsims
Download Presentation

Understanding Relations, Domains, Ranges, and Functions

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Section 1.6 Functions

  2. Section 1.6 Slide 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 3 2 4 1 5 3 4

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

  4. Section 1.6 Slide 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.

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

  6. Section 1.6 Slide 6 Deciding whether an Equation Describes a Function Relation, Domain, Range, and Function Solution 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.

  7. Section 1.6 Slide 7 Deciding whether an Equation Describes a Function Relation, Domain, Range, and Function Example Is the relation a function? Solution • If x = 1, then • Input leads to two outputs: and • Therefore, the relation is not a function

  8. Section 1.6 Slide 8 Deciding whether an Equation Describes a Function Relation, Domain, Range, and Function Example Is the table a function? Solution • 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 • So, the relationis not a function

  9. Section 1.6 Slide 9 Deciding whether a Table is a Function Relation, Domain, Range, and Function Example Is the relation a function? x y 0 2 1 3 1 5 7 3 10 Solution • Input x = 1 leads to two outputs y = 3 and y = 5 • So, the relation is not a function.

  10. Section 1.6 Slide 10 Definition Relation, Domain, Range, and Function Example Is the relation described by the graph a function? Solution • The input x = 1 leads to two outputs: y = –4 and y = 4 • So, the relation is not a function

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

  12. Section 1.6 Slide 12 Deciding whether a Graph Describes a Function Vertical Line Test Solution • All vertical lines intersects the curve at one point

  13. Section 1.6 Slide 13 Deciding whether an Equation Describes a Function Vertical Line Test Example Is the relation y = 2x + 1 a function? Solution • Sketch the graph of • Each vertical line would intersect at just one point • So, the relation is a function

  14. Section 1.6 Slide 14 Definition and Properties Linear Function Definition 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.

  15. Section 1.6 Slide 15 Properties of Linear Functions Linear Function The constant m is the slope of the line, a measure of the line’s steepness. If m > 0, the graph of the function is an increasing line. If m < 0, the graph of the function is a decreasing line. If m = 0, the graph of the function is a horizontal line.

  16. Section 1.6 Slide 16 Properties of Linear Functions Linear Function • If an input increases by 1, then the corresponding output changes by the slope m. • If the run is 1, the rise is the slope m. • The y-intercept of the line is (0, b). • Since a linear equation of the form is a function, we know that each input leads to exactly out output.

  17. Section 1.6 Slide 17 Definition Rule of Four for Functions Definition 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.

  18. Section 1.6 Slide 18 Describing a Function by Using the Rule of Four Rule of Four for Functions Example Is the relation a function? Since is of the form , it is a (linear) function. List some input–output pairs of by using a table. Solution Example

  19. Section 1.6 Slide 19 Describing a Function by Using the Rule of Four Rule of Four for Functions 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 Solution We list five input–output pairs of in the table on the right. Describe the input–output pairs of using a graph. Example

  20. Section 1.6 Slide 20 Describing a Function by Using the Rule of Four Rule of Four for Functions Solution We graph on the right. Describe the input–output pairs of by using words. Example Solution For each input–output pair, the output is 1 less than –2 times the input.

  21. Section 1.6 Slide 21 Finding the Domain and Range Using a Graph to Find the Domain and Range of a Function Example • 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 Solution • No breaks in the graph • Leftmost point: (–4, 2),the rightmost point :(5, –3), • Domain is

  22. Section 1.6 Slide 22 Finding the Domain and Range Using a Graph to Find the Domain and Range of a Function Solution Continued • Range is the set of all y-coordinates of points • Lowest point is (5, –3), highest is (2, 4) • The range is Example Using the graph of the function to determine the function’s domain and range.

  23. Section 1.6 Slide 23 Finding the Domain and Range Using a Graph to Find the Domain and Range of a Function Solution • 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:

More Related