WEEK 11-IDENTIFYING-AND-DEFINING-FUNCTIONS

Identifying and Defining Functions

Objectives

Start

What is and what is not a Function

Definition and Properties

Ordered Pairs and IO Tables

Graphical Representation

Summary

What is and what is not a Function

Before getting started, think about a more general concept: Relationships.

Example 1: Let's consider the parent-child relationship, where each child has assigned a parent:

🗨️

Children

Parents

• Ana

• Peter

• Adriano

• Marie

• Roberto

In this example, there are two children with the same parent, Marie.

But we have no criteria to select, in particular, one parent of Peter.

This is what we DON'T want in a function

Children

Parents

🗨️

• Ana

• Peter

• Adriano

• Marie

• Roberto

Definition and Properties of a Function

Definition: A function is a relationship that assigns exactly one output for each input.

Example 1: Represent in a sagital diagram the function that assigns the numbers x= -2, -1, 0, 1, 2 to the formula

🗨️

2x + 3.

Input x

This means for every x-value (input), there is one and only one y-value (output).

Output y=2x+3

-2-1012

-11357

Characteristics of Functions

Independent (x)

variables

Dependent (y)

How x and y are assigned among themselves.

Definition rule

All possible input values (x-values).

Domain

All possible output values (y-values).

range

Try it by yourself:

Ordered Pairs and Input-Output (IO) Tables

Functions can be represented as sets of ordered pairs (x, y) or tables.

Example 1: Find the ordered pairs of where

Example 2: Write table of values of for

f(x)=x²

f(x)=x + 3.

x=-2, -1, 0, 1, 2.

Dom f={0,1,2,3}

f={(0, 3),(1, 4), (2, 5), (3, 6)}

What is not a Function (Counterexample)

Example 2: Graph the set of pairs and determine whether is a function or not:

Function={(1, 3), (1, 4), (0, 1)}

It is not a function because one input, x=1, has two outputs 3 and 4.

Graphical Representation

A function can be visually represented on a graph.

Example 2: Graph function defined on all of the real numbers.

f(x)=2x+1

Dom f=ℝ

Function of Driving a Car

A function can represent the distance traveled over time. If you drive at 60 km/h, then the distance as a function of time is:

d(t) = 60t

t: time d(t): Distance as a function of time.

a) Find the traveled distance at 0 hours, 1 hour, 2 hours, and 3 hours. b) Plot the points obtained and join them. What geometric shape do you obtain?

07:00

🤔

Summary: Functions

INPUT (X)x=2

INPUT (X)x=-1

ORDERER PAIRS{(2,8), (-1,-1)}

GRAPHIC

f(x)=x³

OUTPUT (Y)Y=-1

OUTPUT (Y)Y=8

Great job!

See you next time

Welcome 6th graders!

A journey soon begin through Social Science experiences!

8TH-IDENTIFYING-AND-DEFINING-FUNCTIONS-EN © 2024 by CASURID is licensed under CC BY-NC-ND 4.0

MATERIAL

It is highly advised to have:

• Grid paper.
• Pencils of different colors.
• Eraser.
• A rule.
• A compass.
• A Protactor.
• A calculator.
• Geogebra installed on your phone/tablet/computer (or use online version).

Solution

Step 1

Step 2

Step 3

Step 4

a)

b)

Step 1

Step 3

Step 4

Step 2

After joining the points, the resulted figure is a straight line (blue).

MA.8.F.1 Define, evaluate and compare functions. MA.8.F.1.1 "Given a set of ordered pairs, a table, a graph or mapping diagram, determine whether the relationship is a function. Identify the domain and range of the relation." MA.K12.MTR.7.1 Apply mathematics to real-world contexts. MA.K12.MTR.5.1 Use patterns and structure to help understand and connect mathematical concepts.