Functions for maths(real functions)

7/29/2019 Functions for maths(real functions)

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The set ofreal numbers is all numbers that can be written on a

number line. It consists of the set of rational numbers and the set

ofirrational numbers.

Irrational numbersRational numbers

Real Numbers

Integers

Whole

numbers


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Recall that rational numbers can be written as the quotient of

two integers (a fraction) or as either terminating or repeating

decimals.

3 = 3.84

5= 0.6

2

31.44 = 1.2


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A repeating decimal may not appear to repeat on acalculator, because calculators show a finite number

of digits.

Caution!

Irrational numberscan be written only as decimals that do

notterminate or repeat. They cannot be written as the ratio of

the pair of integers. If a whole number is not a perfect square,

then its square root is an irrational number. For example, 2 is

not a perfect square, so 2 is irrational.


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Additional Example 1: Classifying Real Numbers

Write all classifications that apply to each number.

5 is a whole number that is not a

perfect square.

5

irrational, real

12.75 is a terminating decimal.12.75

rational, real

16

2whole, integer, rational, real

= = 242

16

2

A.

B.

C.


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Check It Out! Example 1

Write all classifications that apply to each number.

9

whole, integer, rational, real

35.9 is a terminating decimal.35.9

rational, real

81

3whole, integer, rational, real

= = 393

81

3

A.

B.

C.

9 = 3


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A fraction with a denominator of 0 is undefined because you

cannot divide by zero. So it is not a number at all.


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Functions

A function is a relation in which each elementof the domain is paired with exactly one

element of the range. Another way of

saying it is that there is one and only one

output (y) with each input (x).

(x values cannot be repeated).

f(x)x y


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Function Notation

OutputDependent

variable

InputIndependent

variableName ofFunction

y f x


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Domain: a set of first elements in a relation (all

of the x values). These are also called

the independent variable.

Range: The second elements in a relation

(all of the y values). These are also called

the dependent variable.

A relation is a rule that produces one or more

output numbers for every valid input number (x and y values may be repeated).


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Function

X values are always located

on the right and y values are

on the left.

They can be represented by

words, symbols or numbers.

This represents a function as every input value (x)

has only been used once.


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A Relation is a rule that produces one or more

output numbers for every valid input number

(x and y values may be repeated).

This represents only a relation

because the input value or

x-value of 2 was used twice.

Therefore this relation is not aFunction.

All functions are relations but not

all relations are functions!


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Determine whether each relation

is a function.

1. {(2, 3), (3, 0), (5, 2), (4, 3)}

YES, every domain is different!

f(x)

2 3

f(x)3 0

f(x)5 2

f(x)4 3


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Determine whether the relation is

a function.2. {(4, 1), (5, 2), (5, 3), (6, 6), (1, 9)}

f(x)4 1

f(x)5 2

f(x)5 3

f(x)6 6

f(x)1 9

NO,

5 is paired with 2 numbers!


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Is this relation a function?

{(1,3), (2,3), (3,3)}

1. Yes

2. No

Answer Now


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Vertical Line Test (pencil test)

If any vertical line passes through morethan one point of the graph, then that

relation is not a function.

Are these functions?

FUNCTION! FUNCTION! NOPE!


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Is this a graph of a function?

1. Yes

2. No

Answer Now


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Given f(x) = 3x - 2, find:

1) f(3)

2) f(-2)

3(3)-23 7

3(-2)-2-2 -8

= 7

= -8


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Linear Function Definitions

Linear Function= a function that can berepresented by a linear equation.

Linear Equation= an equation that graphs into a

straight line.Standard Form

Ax + By = C

A, B, and C have to be integers


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Example 1

Determine whether the Data in the tablerepresents a linear function.

Step 1

Check the rate of change in the time.

The rate of change is constant, every 10

minutes.

Step 2

Check the rate of change in distance.

This rate of change is NOT constant.

The ANSWER

Since the rate of change is NOT constantfor both variables, the data does NOTrepresent a linear function.

Use the table to find

the solution

Time

(min)

Distance

biked

(miles)

10 3

20 6

30 10

40 14

50 17

60 19

+10

+10

+10

+10

+10

+3

+4

+4

+3

+2


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Review: Make a t-table

If f(x) = 2x + 4, complete a table

using the domain {-2, -1, 0, 1, 2}.

2(-2) + 4 = 0 (-2, 0)2(-1) + 4 = 2 (-1, 2)

2(0) + 4 = 4 (0, 4)2(1) + 4 = 6 (1, 6)

2(2) + 4 = 8 (2, 8)

x f(x)

-2-1

01

2

ordered pair


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Given the domain {-2, -1, 0, 1, 2},

graph 3x + y = 6

-3(-2) + 6 = 12 (-2, 12)

-3(-1) + 6 = 9 (-1, 9)-3(0) + 6 = 6 (0, 6)

-3(1) + 6 = 3 (1, 3)

-3(2) + 6 = 0 (2, 0)

x -3x + 6 ordered pair

1. Solve for y: 3x + y = 6

Subtract 3x - 3x - 3x

y = -3x + 62. Make a table

-2

-10

1

2


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Bonus questions!

What is the x-intercept?

(2, 0)

What is the y-intercept?

(0, 6)

Does the line increase or decrease?

Decrease

5) Given the domain {-2, -1, 0, 1, 2},

graph 3x + y = 63. Plot the points

(-2,12), (-1,9), (0,6), (1,3),

(2,0)

4. Connect the points.


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3-2 The Slope of a Line

Finding the slope of a line given two pointson the line:The slope of the line through two distinct points

(x1, y1) and (x2, y2) is:

Note: Be careful to subtract the y-values and the x-values in thesame order.

Incorrect

12 1

2 1

2 ( x )x

y yrise change in yslope mrun change in x x

x

2 1 1 2

1 2 2 1

y y y y

or

x x x x


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3-2 The Slope of a Line

Finding the slope of a line given two points onthe line:

Example 1) Find the slope of the line through the points

(2,-1) and (-5,3)

2 1

2 1

3 ( 1) 4 4

= ( 5) 2 7 7

y yrise

slope m run x x


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2) 2x2 - y = 7

Can you write it in standard form?

NO - it has an exponent!Not linear

3) x = 12

x + 0y = 12

A = 1, B = 0, C = 12

Linear

Determine whether each equation is

a linear equation.


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Heres the cheat sheet! An equation that is

linear does NOT contain the following:1. Variables in the denominator

2. Variables with exponents

3. Variables multiplied with othervariables.

xy = 12

32y

x

2 3y x


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Lets examine exponential functions. They aredifferent than any of the other types of functions weve

studied because the independent variable is in the

exponent.

xxf 2

Lets look at the graph of

this function by plotting

some points.x 2x

3 82 41 2

0 1-1 1/2-2 1/4-3 1/8

2-7-6-5-4-3-2 -1 1 5 730 4 6 8

7

1

23456

8

-2-3-4-5-6

-7

2

121 1 f

Recall what anegative exponent

means:

BASE


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xxf 2

xxf 3

Compare the graphs 2x, 3x , and 4x

Characteristics about the

Graph of an Exponential

Function where a > 1 xaxf

What is the

domain of an

exponentialfunction?

1. Domain is all real numbers

xxf 4

What is the range

of an exponential

function?

2. Range is positive real numbers

What is the xintercept of these

exponentialfunctions?

3. There are no x intercepts becausethere is no x value that you can putin the function to make it = 0

What is the yintercept of these

exponentialfunctions?

4. The y intercept is always (0,1)because a 0 = 1

5. The graph is always increasing

Are these

exponential

functionsincreasing or

decreasing?

6. The x-axis (where y = 0) is a

horizontal asymptote forx

-

Can you see the

horizontal

asymptote forthese functions?


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All of the transformations that you

learned apply to all functions, so what

would the graph of

look like?

xy 232 xy

up 3

xy 21

up 1Reflected over

x axis 122 xy

down 1right 2


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30

PROPERTIES

?0

1

0

1

0

aa

a


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31

PROPERTIESan am an+m

am : an am-n

(am)n amn

n mn

m

n

n

aa

aa

1


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Indicate if every raltionship is true or false:

1. 53 = 15 T F

2. 24 = 16 T F

3. T F

4. (53)7 = 521 T F

5. 00 =1 T F6. 83 : 83 = 0 T F

7. (73 : 72)0 = 1 T F

8. (142 : 72) = (14 : 7)2 T F

9. (15

3

: 3

3

) = 125 T F10. (24)3 : (24)2 = 16 T F

11. (34)2 3 = 39 T F

12. 23 25 : 22 : 28 = 2-2 T F

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Functions for maths(real functions)