2.4 Relations Graphs and Functions

Lesson Plan 2.4: Relations, Graphs, and Functions

Objectives:

Students will be able to:

• Define and identify sets. (collections of objects)
• Define and identify relations and functions.
• Define and identify domain and range of relations.
• Work with function notation.

Suggested Lecture Breaks:

• MWF: You have two days. The first day, get through Problem 4.
• MW/TR: You have 1.5 days. The first day, get through Problem 5.

Suggested Lecture Notes:

Relations and Functions

• Give definition of function; draw pictures to explain input-output relationship.
• Have students work on Problem 1, gather consensus.
• Provide examples of relations and ask students whether the relations represent functions or not.
• Represent a finite relation in several ways.
• Have students represent the finite relations in Problem 2 in different ways, for example turn 2.a into an arrow-diagram or turn 2.c into a set of ordered pairs.
• Define domain and range of a relation.
• Ask a student to convert 4a. into an arrow diagram.
• Introduce real valued functions of real variables (maybe don't call them that) and the vertical line test, direct attention to problem 4.
• Include more examples of graphs and ask students if they are functions.
• Explain how real valued functions can be given by formulas (equations).
• Have students work on problems 1-4.

Function Notation

• Define function notation; discuss how to read function notation and input vs. output.
• Make sure to include a discussion on how to interpret the terminology "y is a function of x" versus "x is a function of y". It may be helpful to provide a simple example of this to illustrate the difference. For example: say you have data on how many people visited the zoo based on the average temperature of that day:

Average Temp in Degrees F	# of zoo visits
60	2300
78	5600
85	3600
78	1500

Ask: Is average temperature a function of the # of zoo visits? Is the # of zoo visits a function of average temperature?

• Have students work on Problem 5.

• Provide examples of functions, such as:
  • f(x)=3x+2
  • g(x)=x^2-1
  • Choose points to evaluate at. For example: x=2, x=3^2, x=1/2, x=x+h (this one will be trickier, but it's important to get them used to working with inputs that aren't just real numbers. It may be helpful to say in words, where the input is "x+h").
• Have students work on problems 6-10.

Comments on the handout

• When converting table e. to the other kinds of diagrams, students might have trouble with the repeated input. Perhaps show that the tables and sets of ordered pairs are more similar to eachother than they are to the arrow diagrams.
• There are several more kinds of graphs that could be used when talking about real valued functions in problem 4. Silly shapes, circles, logos, etc. These examples show strength of the vertical line test.
• Problems 4 d, e, and f are likely to pose a challenge.