01 Functions

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Unit 8

Functions & GraphsF2N

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Definition of a Relation

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Definition of a Relation

A Relationmaps values from one subsetto the of values another subset. ARelationis a set of ordered pairs.

The most common types of relations inalgebra map subsets of real numbers toother subsets of real numbers.

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Mapping diagram illustrates how each member of

the domain is related with each member of the range

x y

04

5

7

-91

2

Example: Draw a mapping for the following.(5, 1), (7, 2), (4, -9), (0, 2)

(Note: First list values of x and y once each, inorder.)

How to show relation?

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5

6

7

8

9

1

2

3

4

5

The Rule is ADD 4 to the domain

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Ahmed

Peter

Ali

Jaweria

Hamad

Paris

London

Dubai

New York

Cyprus

Has Visited

There are MANYarrows from each person and each place is related to MANYPeople. It is a MANY to MANYrelation.

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Bilal

Peter

Salma

Alaa

George

Aziz

62

64

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Person Has A Mass of Kg

In this case each person has only one mass, yet several people have the sameMass. This is a MANY to ONErelationship

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Is the length of

14

30

Pen

Pencil

Ruler

Needle

Stick

cm object

Here one amount is the length of many objects.This is a ONE to MANYrelationship

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FUNCTIONS

Many to One Relationship

One to One Relationship

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Only involve 2 kinds of relationship:-

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Domain Co-domain

0

1

2

3

4

12

345678

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Image Set (Range)

x2x+1A B

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:x

x2

4

x x2 4

The upper function is read as follows:-

Function fsuch that xis mapped onto x2+4

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Lets look at some functionType questions

Iffxx2 4andgx1-x2

Findf2

Findg3

x x2 42 2 = 8 gx 1 -x2

3 3

= -8

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Consider the functionfx 3x - 1 We can consider this as two simplerfunctions illustrated as a flow diagram

Multiply by 3 Subtract 13x 3x - 1x

Consider the functionf:x2x 52

xMultiply by 2 Add 5

2x 2x 5Square

all

2x 52

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:x3x 2 and gx :xx2Consider 2 functions

g is a composite function, where gis performed first and then fis performedon the result of g.The function fgmay be found using a flow diagram

xsquare

x2Multiply by 3

3x2Add 2

3x2 2

Thus = 3x2 2

f

f

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x

2 3x 2

3x2 2

24 14

2

f

f =15


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Consider the function x 5x - 2

3

Here is its flow diagram

x

x5x5 -2

5x - 23

Draw a new flow diagram in reverse!. Start from the right and go left

Multiply by 5 Subtract 2 Divide by three

Multiply by threeAdd twoDivide by 5

x3x3x +23x +2

5

-1x 3x 2

5And so

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Example

Find the domain and range of the relation.

{(5,12), (10, 4), (15, 6), (-2, 4), (2, 8 )}

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Example

Determine whether each relation is a function:

A) {(1,2), (3,4), (5,6), (5,8)}

B) {(1,2), (3,4), (6,5), (8,5)}

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Functions as Equations

a) b)

1. Solve for yin terms of x.

2. If two or more values of ycan be obtained

for a given x, the equation is not a function.

44222 yxyx

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Solve for y and determine if theequation is a function.

A) 2x + y = 6 B) x2 + y2 = 1

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Evaluating a Function

Common notation: f(x) = function

Evaluate the function at various values of x,represented as: f(a), f(b), etc.

Example: f(x) = 3x 7Then, f(2) =

f(3 x) =

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If f(x) = x2 2x + 7, evaluate each of the

following.

a) f(-5) b) f(x + 4) c) f(-x)

Ans: a) 42

b) x2+6x + 15

c) x2 + 2x + 7

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Determine if a relation is a functionfrom the graph?

Remember: to be a function, an x-value isassigned to ONLY one y-value .

On a graph, if the x value is paired with MOREthan one y value there would be two pointsdirectly on a vertical line.

THUS, the vertical line test! If a vertical linedrawn on any part of your graph touches morethan one point, it is NOT the graph of afunction.

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To determine Graphs ofFunctions

Step 1: Graph the relation. (Use graphingcalculator or pencil and paper.)

Step 2: Use the vertical line test to see if therelation is a function.

Vertical line test If any vertical line

passes through more than one point ofthe graph, the relation is not a function.

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(a)

(b)

(c) (d)

(a) and (c)25


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Determine if the graph is a function.

a) b)y

x

5

5

-5

-5

y

x

5

5

-5

-5

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Heres more practice.

c) d)y

5

5

-5

-5

y

x

5

5

-5

-5

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Can you identify domain & range fromthe graph?

Look horizontally. What x-values are containedin the graph? Thats your domain!

Look vertically. What y-values are contained inthe graph? Thats your range!

Write domain and range using interval or set-builder notation.

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What is the domain & range of thefunction with this graph?

) : ( , ), : ( , )

) : ( 3, ), : ( , )

) : ( 3, ), :( 3, )

) : ( , ), : ( 3, )

a Domain Rangeb Domain Range

c Domain Range

d Domain Range

- - - -

- -

- -

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Additional Example

Graph the function.

Then estimate thedomain and range.

( ) 1f x x -

( ) 1f x x -

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Finding intercepts:

x-intercept: where the function crosses thex-axis. What is true of every point on thex-axis? The y-value is ALWAYS zero.

y-intercept: where the function crosses they-axis. What is true of every point on they-axis? The x-value is ALWAYS zero.

Can the x-intercept and the y-interceptever be the same point? YES, if thefunction crosses through the origin!

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Summary

Domain = x values

Range = y values

Use the vertical line test to verify if a graphis a function.

To evaluate means to substitute and

simplify. Intercepts where function crosses the x-

or y-axis

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01 Functions