Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a...

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Transcript of Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a...

Page 1: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.
Page 2: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.

•Functions are used to model mathematical situations.

Examples:A=πr2 The area of a circle is a

function of its radiusC=5/9(F – 32) °C as a function of

degrees °F

Functions: What the f?

Page 3: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.

Function notation

y = 1 - x2 f(x) = 1 - x2

“y is a function of x” function notation

f(x) is read “f of x”

Page 4: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.

Evaluating functions

f(x) = 1 - x2

What is f(-1)?In other words, evaluate the function

f(x) at x = -1.f(-1) = 1 – (-1)2 = 1 – 1 = 0

Page 5: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.
Page 6: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.

Domain: For what values of x is the function defined?

Range: For what values of y is the function defined?

Page 7: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.

Basic examples

f(x)=ax2+bx+c {all real numbers} g(x)=1/x {x: x 0} *denominator cannot be 0

h(x)=√x {x: x ≥ 0}

v(x)=ln(x) {x: x > 0}

Page 8: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.

**** Notation time-out******

In mathematics,: means “such that”

Ex: {x : x 0} means “the set of all x’s such that x

0”

Page 9: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.

Domain of

y = 2x + 3

{x: 2x + 3 0}

“all x’s such that 2x+3 is greater than or equal to 0”

Page 10: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.

Domain of

y =

{x: 2x + 3 0}

Or {x: x -3/2}

1 2x +

3

Page 11: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.

Domain of

y = 2x + 3All real numbers

3

Page 12: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.

Domain of

y = ln(2x + 3)

{ x: 2x + 3 > 0 }

Page 13: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.

Domain of

y =

{ x: 2x + 3 > 0 }

1 2x + 3

Page 14: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.

Assignment Ap.27/ 1,2,(3-23)odd,37

Page 15: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.

SAT Problem of the Day!

“700 on the SAT math!!! Heck yes, I am going to UCLA!”

Page 16: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.

* A more CONCISE way to describe sets.

** Is used interchangeably with set notation to express domain and range.

Page 17: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.

{x:…} INTERVAL

x 3

x 3

x 3

x 3

x 3

(- , 3)

(- , 3]

(3, )

[3, )

(- , 3) & (3, )

Page 18: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.

-2 x 3

-2 x 3

-2 x 3

-2 x 3

All reals (- , )

(- 2 , 3][- 2 , 3)

(- 2 , 3)

[- 2 , 3]

Page 19: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.

MIX AND MATCH!!!

Page 20: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.

Assignment BIgnore the directions. Instead…Find the domain of each function

and write it using a) set notation and b) interval notation.

p.27-28/ 4,6,8,20,30,34

Page 21: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.

SAT Problem of the Day!

Page 22: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.
Page 23: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.

y = |x|

Absolutevalue

Page 24: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.

y = x2

parabola

Page 25: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.

y = x

Evenroot

Page 26: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.

y = x3

Oddroot

Page 27: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.

1

y = e x

Exponentialgrowth

Page 28: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.

1

y = e - x

Exponentialdecay

Page 29: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.

1

y = ln x

Page 30: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.

y = 1/x

Page 31: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.

y = 1/x2

Page 32: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.

y = x2 - 2

y = x2

y = x2 + 3

+ move up - move

down

Page 33: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.

y = |x|

y = |x - 6| y = |x +3|

+ move left - move

right

Page 34: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.

y = log 2 x

y = log 2 (x + 2)

left2

Page 35: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.

y = x

y = - x Flip about x

Page 36: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.

1

y = e x

y = - e x

flip

Page 37: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.

1

y = - ln x

Page 38: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.

1

y = - e -x

Page 39: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.

Given the graph of y = f(x),

To graph y = f(x) ± a,

Move the graph of y = f(x) up/down a

units

Page 40: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.

Given the graph of y = f(x),

To graph y = f(x ± a),

Move the graph of y = f(x) left/rt a units + is left, - is right!

Page 41: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.

Given the graph of y = f(x),

To graph y = -f(x),

flip the graph of y = f(x) with respect

to the x-axis.

Page 42: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.

The graph of y = - f(x) is

flipped about the x-axis.

The graph of y = f(-x) is

flipped about the y-axis.

Page 43: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.

y = -x

Page 44: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.

1y = e -

x

Page 45: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.

1

y = ln (-x)

Page 46: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.

Given the graph of y = f(x),

To graph y = kf(x),

Multiply all the y values of y = f(x) by k. Steeper if k > 1.

Flatter if k < 1

Page 47: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.

y = x

y = 2x steeper

Page 48: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.

y = |x|

y = 2|x|

y = ½ |x|

Page 49: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.

y = x3

y = 2 x 3

double they-values

Page 50: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.

y = x3

y = ½ x 3

half they-values

Page 51: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.

1

y = e x

y = 2 e x

2

Page 52: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.

Assignment Cp. 28/ 47-55,83,93

Page 53: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.

SAT Problem of the Day!

Page 54: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.
Page 55: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.

y = |x|

y = -|x - 4| + 3

flip right 4up 3

y = -|x - 4| + 3

Page 56: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.

y = |x|

y = -2|x - 4| + 3 flip right 4

up 3

y = -2|x - 4| + 3

steeper

Page 57: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.

y = x2

y = -(x - 6)2 + 1

flip right 6 up 1

y = -(x - 6)2 + 1

Page 58: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.

SAT Problem of the Day

Page 59: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.
Page 60: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.

(f g)(x) f (g(x))

Given two functions y=f(x) and y=g(x).

“f composed with g of x”

Page 61: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.

a) (f g)(x)

Example: Finding composite functions.

Given f(x)=2x+3 and g(x)=cos(x). Find…

f (g(x))

2(cos(x)) 3

b) (g f )(x)

g(f (x))

cos(2x 3)

*Usually, (f g)(x) (g f )(x)

2cos(x) 3

Page 62: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.

Example: Finding composite functions.

Given f(x)=2x+3 and g(x)=cos(x).

d) What is (g f )( 2)?(g f )( 2)

g(f ( 2))

g(2 2 3)

g( 1) 0.54

c) What is (f g)( )?(f g)( )

f (g( ))

f (cos( ))

f ( 1) 1

Page 63: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.

Try this…

Find two functions f and g such that F(x)= (f g)(x)

a) F(x) 3x 6

2b) F(x) ln(x 4x 4)

Page 64: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.
Page 65: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.

Even vs. Odd

ODD functions are symmetric with respect to the origin.

EVEN functions are symmetric with respect to the y-axis.

Page 66: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.

Even or Odd?

Page 67: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.

Even or Odd?

Page 68: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.

Even or Odd?

Page 69: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.

Even or Odd?

Page 70: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.

Even or Odd?

Page 71: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.

Even or Odd?

Page 72: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.

Even and Odd FunctionsEven, odd, or neither test:The function y=f(x) is EVEN if f(-x) = f(x).

The function y=f(x) is ODD if f(-x) = -f(x).

Otherwise, it is neither even nor odd.

Page 73: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.

ExampleDetermine whether the function is even,

odd, or neither.a) b) 3f (x) x x g(x) 1 cos(x)

3f ( x) ( x) ( x) 3x x

f (x)

odd

g( x) 1 cos( x) 1 cos(x) * cos is even

g(x)

even

Page 74: Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function.

Assignment E

p. 28-29/ 57 – 70

*Test on Unit P: FunctionsThis Thursday! Review Assignment onlinewww.geocities.com/mskadlac