1.1 Functions

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Objectives:

• Determine when a relationship is a function and determine its inputs and outputs
• Represent relations/functions as tables, ordered pairs, graphs, equations, etc.
• Review graphs and the vertical line test

Definitions

function

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Lesson Guide

Determine when a relationship is a function and determine its inputs and outputs

[15 minutes] Begin this lesson by giving examples of relationships between two sets. Examples should be easy to understand. I.e., use real world examples such as the days of the year and their average daily temperatures or the students in your class and the month they were born. Discuss how we can identify objects in the two sets and call this identification a “relation.”

A lot of this class will be about exploring the relationship between different pieces of  information. For example:

Instructor Example

Suggested Example

This is a sentence that is in place to generate space in the table

The year you were born is related to your age. The month is related to the temperature outside. The price of gold is related to the supply and the demand. If we have a very special kind of relationship called a ``function," we can use one piece of information, called the input, to completely predict the other piece of information, called the output. It is often useful to take a few minutes and draw bubble diagrams of the examples you are going over. E.g.

You'll now want to define what a function is. Start by using the terms input and output to describe bubbles in your bubble diagram. Say the definition twice for emphasis.

A function is a relation (or a rule) that assigns each input to only one output.

We are not introducing function notation just yet, that comes shortly. Stress the importance of using and understanding precise language. Tell your students that if they are ever asked if a relationship is a function, they have to show that the relation either fulfills or violates the definition of a function.

Instructor Example

Suggested Example

This is a sentence that is in place to generate space in the table

Is the number of letters a function of a word? First, identify what is the input and what is the output. For example: Input: Word Output: Number of Letters I 1 Love 4 Math 4 Class 5 In this case, the number of letters is a function of the word, since for each input (word) has exactly one output (number of letters). Students may notice that there are some words that have multiple accepted spellings (e.g. color and colour). Let the discussion progress naturally; have them consider whether it is still a function. For example, you may institute the rule that words with different spellings are different words, or else agree to some master list of the proper way to spell each word. Use this conversation to emphasize being precise and careful with language.