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• PowerPoint Slide Show (.pps)• You can advance through each part of the screen by

left clicking• When you see the at the top right of the slide

you can click it to advance to the next slide.

Introduction

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This slideshow was developed for those students who want to explore more about Senior 2 Math Applied Functions and Relations. When done you should be able to:– Plot linear and non-liner data using appropriate scales

– Represent data using function models

– Use a graphing tool to draw graphs of a function

– Describe a function as ordered pairs, a rule, or in words

– Use function notation to evaluate and represent functions

– Determine the range and domain of a function or relation

Functions and RelationsPrepared by Mr. F

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•Relation: A rule that produces one or more output numbers for every valid input number.

Definition: Functions and Relations

• Function: A rule that gives a single output number for each input number.

Time (t) Distance (d)

10 minutes 6 km

30 minutes 6 km

40 minutes 12 km

Input numbers Output Numbers

x y1 y2

0 2 -2

1 1 -1

2 0 0

Input numbers Output Numbers

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Example of a Function

From the previous example: Time (t) Distance (d)

10 minutes 6 km

30 minutes 6 km

40 minutes 12 km

So Susie drives the car to the library and then returns home. The table and the graph show the same thing: how many km Susie put on the car for each point of time.

Notice there is only one distance for every time (t), so this is a function. (Can you be two places at the same time??)

Susie arrives at library

Susie leaves library

Distance Susie has put on car

when she arrives home

This is a function since there is only one output number for each input number

Input Number Output Number

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Example of a Relation

Here is an example of a relationX Y1 Y2-4 0.0 0.0-3 2.6 -2.6-2 3.5 -3.5-1 3.9 -3.90 4.0 -4.01 3.9 -3.92 3.5 -3.53 2.6 -2.64 0.0 0.0

Notice each input value, (x), has two output values (y). So for example: an input of 2 can have an output of 3.5 or –3.5

So this is a relation

There are many values of x that have two y values in this circle

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Function or Relation?

How can you tell what is a function and what is a relation?– By the definitions:

• Function: A rule that gives a single output number for each input number.

• Relation: A rule that produces one or more output numbers for every valid input number.

– Also:

• The graph vertical line test. If you can draw a vertical line through the graph and there are more than two y values then it is a relation

• The y-power test. If the y has an an even power. Example y2 or y4

• The ordered pairs test. In ordered pairs or a table, if there exists more than one output value (y) value for each input value (x) anywhere in the data.

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Function vs Relation Quiz 1

Is each of these a function or a relation?

x

y

x

y

x

y

x

y

x

y

x

y

function function

function

relation

relationrelationIf you need to review why these are the answers click on this block

There are 2 output values of y for most values of x

There are 2 output values of y for most values of x

There are many output values of y for some values of x

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Function vs Relation Quiz 2

What do these sets of ordered pair points represent:

a function or a relation?• {(1,1) (1, 3) (2,5) (3,6)}• {(1,2) (2,3) (3,5) (5,2)}• {(0,1) (1,5) (2,5) (2,9)}• {(9,4) (10,13) (20,25) (21,25)}

Relation!

Relation!

Function!

Function!

Value of 1 has two outputs

Value of 2 has two outputs

If you need to review why these are the answers click on this block

If you the last two pages completely right then click on this box

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Even Power Test of an Equation

• If b2 =4; what is the value of b? That is: what number, b, multiplied by itself gives 4?

Did you say 2?

You are only half right because –2 works also!So if we said y2=x2. How could you figure

out what y to graph for each x?

To graph it you want to get y by itself so you would say:2xy

Or y = plus and minus the square root of x2. So y= +x and –x. There are two values of y for each x, ( x and –x), so the equation y2=x2 is a relation

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y2 = x2 Relation

• Let’s look at the very simple equation and relation y2 = x2. A table of the solution would look like this

x y1 y2

-2 -2 +2

2 2 -2

4 4 -4

6 6 -6

The graph looks like this

Notice there is more than one value of y for some of the x

Notice also, that a vertical line passing through the graph will touch the curve at more than one place

So y2 = x2 is a relation

y

x

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Graphing a relation on the TI 83

• You already know how to graph functions using the Y= button on the TI 83.

• We already saw that a relation has two y values for at least one x value

• So to graph a relation on the TI 83 we need to break into two parts.

• Let’s try y2=x. In other words y=+ and – the square root of x.

xy

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Graphing a relation : y2 = x

If y2 = x, then xy So there are really two graphs to do:

so press: [y=], cursor to Y1 in the equation editor screen, press [2nd], [x]

Now enter the other half for the negative values of values of y into Y2:

so press: [y=], cursor to Y2 in the equation editor screen, press [(-)], [1], [*], [2nd], [x]

Press [GRAPH]The graph should look like this:

If you want to try graphing more relations try this Exercise

(Will open in Word 200)

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Function Notation

• Mathematicians often use a special ‘notation’ to represent functions, they say there exists a function f(x) (A function is different from a relation remember)

• f(x) is pronounced ‘f at x’• f(x) means: what is the output value associated with

a function ‘f’ that works on x. In this idea a function ‘f’ is like a machine

f(x)Something goes in (input)

Something comes out (output)

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Function Machines you Already Know

You already know lots of functions and function machines!– You know the machine f(x) = x2

– You know the machine f(x)= SIN (x)

So f(x) just means you feed something into a function machine and something comes out!

F(x) is just a ‘machine’ that converts one number into one other! It is a function machine!

f(x)=x2x goes in (input) x2 comes out (output)

f(x)=sin (x)x goes in (input) Sin (x) comes out (output)

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30° 0.5000

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Guess the Function Machine - 1!

Here is a function machine f(x), no! wait! we will call it g(x). (Who cares what you call it anyway!) Guess what the machine does!

In g(x) Out

2 g(x) 3

3 g(x) 4

4 g(x) 5

40 g(x) 41

100 g(x) 101

Click anywhere for the answer

If you guessed that the g(x) machine just added one to the input number then you were right! Or we might say ‘out = in plus 1’. Or maybe even y=x+1 if you want to call in ‘x’ and out ‘y’

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Evaluating a Function

• To Evaluate a function means to find the output value for a particular input value.

f(x)=x22 goes in (input) 4 comes out (output)

• To Evaluate a function we just ‘plug in’ the value of x into the function.

• Click to try these two simple Evaluation problems

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So what does ‘f’ equal?

Do not get caught out! f(x) is just the ‘notation’ for a function ‘machine’. – It isn’t f * x.!!! Or f times x– It is just notation; a way that math folks represent that ‘there exists’ a

function or a ‘machine’ if you want.

Mathematical notation is just the way we represent ideas economically. The idea of f(x) is that there exists a function that does something to an input number to spit out a single output number

Notation is just like the language of math! It is the meaning that counts! So notation is important. If you say f(x) to a Chinese math student on the internet he will know exactly what you mean because it is the globally common ‘notation’ or language.

‘f’ doesn’t mean anything!!!!!!!!!!. It is f(x), pronounced “f at x”, that represents the important idea. The fact that there exists a function, f(x), or a machine that takes something (an input number) and converts it into something else (an output number)! A simple idea when you think about it.

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Domain and Range

• You are already familiar with the idea of domain and range

• Definitions– Domain of a function or relation: The set of all possible x-

values (input values) (or valid input types) represented by a graph or an equation

– Range of a function or relation. The set of output numbers of a function or a relation.

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What does Domain and Range really Mean??

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