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Section 1.1 Functions

A function f from D into R is a rule that assigns to each element x D a unique element y R.

The set D is called the domain of f and the set of image of D under f is called the range of f.

1.5 domain range eve odd functions

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b.

a.

c.

.x

.y

.z

P Q

f

Is f a function? YES! Domain of f : { a, b, c}

Range of f : {x, y, z}

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b.

a.

c.

.x

.y

.z

P Q

g

Is g a function? YES! Domain of g: { a, b, c}

Range of g: {x, y}

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a.

c.

.x

.y

.z

P Q

h

Is h a function? YES! Domain of h: { a, c}

Range of h: {x, y}

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b.

a.

c.

.x

.y

.z

P Q

i

Is i a function? NO! i is not a function because ……..

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How graph of a function supposed to be?

Vertical line test: A curve in a plane is a graph of a function if no vertical line intersects the curve in more than one point.

Yes! This is a graph of a function because…

No! This is not a graph of a function because…

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Vertical Line Test

• A curve in the xy-plane is the graph of a function of x if and only if no vertical line intersects the curve more than once

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Determine whether the following are graphs of functions

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x=t^2-2, y=5-2t, A parabola

-8 -6 -4 -2 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

-10

-5

5

10

15

20

25

x

f(x)

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r(t)=cos (2t)

-1.2 -1 -0.8 -0.6 -0.4 -0.2 0.2 0.4 0.6 0.8 1 1.2

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0.2

0.4

0.6

0.8

1

x

f(x)

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r(t)=sin ((8t)/5)

-1.2 -1 -0.8 -0.6 -0.4 -0.2 0.2 0.4 0.6 0.8 1 1.2

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0.2

0.4

0.6

0.8

1

x

f(x)

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f(x)=2cos(x)+sin(2x)

-11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8

-10

-8

-6

-4

-2

2

4

6

8

x

f(x)

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r(t)=cos(4t) A rose with eight leaves

-0.8 -0.6 -0.4 -0.2 0.2 0.4 0.6 0.8 1 1.2 1.4

-1

-0.8

-0.6

-0.4

-0.2

0.2

0.4

0.6

0.8

1

1.2

x

f(x)

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r(t)=2sqrt(cos(2t)) Lemniscate

-2.5 -2 -1.5 -1 -0.5 0.5 1 1.5 2

-2.5

-2

-1.5

-1

-0.5

0.5

1

1.5

2

x

f(x)

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f(x)=1/(x̂ 2+2x+2)

-3.5 -3 -2.5 -2 -1.5 -1 -0.5 0.5 1

-2

-1.5

-1

-0.5

0.5

1

1.5

2

2.5

x

f(x)

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r(t)=(e^cos(t))-2(cos(4t))+(sin(t/ 4))^3 Butterfly !!

-3 -2.5 -2 -1.5 -1 -0.5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5

-4

-3

-2

-1

1

2

3

4

5

x

f(x)

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f(x)=0.6x+24cos(0.4x)+0.2x

-25 -20 -15 -10 -5 5 10 15 20 25 30 35 40 45 50

-35

-30

-25

-20

-15

-10

-5

5

10

15

20

25

30

35

40

x

f(x)

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x(t)=sin(3t) , y(t)=sin(4t)

-1 -0.8 -0.6 -0.4 -0.2 0.2 0.4 0.6 0.8 1 1.2

-1

-0.8

-0.6

-0.4

-0.2

0.2

0.4

0.6

0.8

1

1.2

x

f(x)

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x(t)=t +sin(2t) , y(t)=t+sin(3t)

-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

-8

-6

-4

-2

2

4

6

8

x

f(x)

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Domain & Range

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Linear Functions

Domain of P ?

Range of P ?

{ x (-, + ) } = R

{ y (-, + ) } = R

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Quadratic function

Domain of f : { x (-, + ) } = R

Range of f : { y [0, +) }

f

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f(x)=x̂ 3

-2 -1.5 -1 -0.5 0.5 1 1.5 2

-2.5

-2

-1.5

-1

-0.5

0.5

1

1.5

2

x

f(x)

Domain of f ?

Cubic function

{ x (-, + ) } = R

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What can we conclude about the domain of polynomials?

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Domain & range

Domain of f : { x - < x 0 } Range of f : { y 0 y < }

f

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Domain of f ?

Range of f ?

f

Rational Functions

)},0()0,(|{ x

)},0()0,(|{ y

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Rational Functions

Domain of R?

Range of R?

)},2()2,(|{ x

)},0(|{ y

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Question: how to find the domain of a function without looking at the graph?

Ex 1: Find the domain of .9

1)(

2

xxf

Df= All real numbers except -3 & 3 or can be written as Df= { x (-, -3) (-3, 3) (3,+)}.

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Ex 3: Find the domain of .4

5)(

2

xxf

Ex 2: Find the domain of .3)( xxg

Solution :Ex 2 : ),3[

Ex 3 : ),2()2,(

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Even & Odd functions

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even & odd functionsIf a function f satisfies f(-x) = f(x) for every number x in it’s domain, then f is called an even function.

If a function f satisfies f(-x)= -f(x) for every number x in it’s domain, then f is called an odd function.

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Are the following functions even, odd or neither? f(x) = x2, g(x) = x3,

h(x) = - x2 + 4x – 3

Check it by doing some calculation!

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Just by looking at the graph, how can we tell that the function is even,

odd or neither?

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EVEN FUNCTION

NOTICE THAT f(-x) = f(x)f IS SYMMETRY ABOUT Y-AXIS

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ODD FUNCTION

NOTICE THAT f(-x) = -f(x)f IS SYMMETRY ABOUT ORIGIN