Where is the vertex of this graph?

142 xxy

5)2( 2 xy

)5,2(

Domain & Range

• Know what the “domain” of a function is. Know what the “range” of a function is.

• Understand how to describe the domain and range of functions and the meaning of notation like xєR 2<x<8

• Be able to explain how the range and domains of functions can be limited and give examples.

Functions

xf

What is a function, and what does it do?

Functions

12 xxf

What is a function, and what does it do?Functions are like machines, you put a input in, and you get a output.

31 f

is a image of x xf

12: xxf Notation: 12 xxf

Functions 12 xxf

11

10

12

5

0

2

Functions are sometimes called mappings. It maps x values to an output.

x 12: xxf

Domain & RangeSometimes functions have restrictions. To fully describe a function, you need to define its domain and range.

The domain of a function is given by the set values of x or inputs

22,: 2 xxxf

xxxf ,: 2 ℝ

ℝ means real numbers

What is the Domain?

• Domain: xєlR -5≤x<7

What is the Domain?

• Domain: xєlR:-1<x≤1

What is the Domain?

• Domain: xєlR -5<x<-3

What is the Domain?

• Domain: xєlR -6≤x≤6

What is the Domain?

• Domain: xєlR -1≤x<11

Infinite Domains

• Domain: xєlR:x≥-5

Infinite Domains

• Domain: xєlR:x>2

Infinite Domains

• Domain: xєlR (x can take any value)

Function Domains

• Domain: xєlR x≥0

Even roots cannot be done on negative numbers

Function Domains

• Domain: xєlR x≠0

You cannot divide by 0(note also that the output cannot be 0 you’ll need this later so don’t forget)

Domain Round Up

• The domain of a function is a description of the values of x for which it is to be defined.

• There are various reasons that a function may only take limited values.– It may be impossible to evaluate the function for

particular values of x.– It may be that we are only interested in certain

values (e.g. If the value of x represents time it cannot be negative)

Range of f(x) = x2 is

Domain & RangeRange of a function is the set values given by xf

2xx

0y

If y = f(x), then the range is given by y-values

Range of f(x) = x2 is 22,2 xx

40 xf

Domain – set x values, Range – set y values

Alphabetically D is before R, and X is before Y

,y ϵ ℝ

What is the Range?

• Range: yєlR 0≤y<25

–4 –3 –2 –1 1 2 3 4 5 6

5

10

15

20

25

30

x

y

–2 –1 1

–5

–4

–3

–2

–1

1

2

3

4

x

y

What is the Range?

• Range: yєlR -2≤y≤2

–1 1 2–1

1

2

3

4

5

6

7

8

x

y

What is the Range?

• Range: yєlR y>1/5

1 2 3 4 5 6 7

1

2

3

4

5

6

7

x

y

What is the Range?

• Range: yєlR y≥3

All quadratics have a range that is greater or equal to their minimum y value (or less than or equal to their maximum for –ax2)

Range Round Up• The range of a function is a description of the

values of y (outputs) for which it is defined.• There are various reasons that a function

output may only take limited values.– It may be impossible to have particular values

come out e.g.² will never be negative.– It may be that we are only interested in certain

values of x therefore limiting how many come out.

Domain – set x values, Range – set y values

Alphabetically D is before R, and X is before Y

Domain & Range

Domain – X values

Range – Y values

Domain & Range

The domain of the kraang is in Dimension-X

Domain = x values

Domain & Range of QuadraticsQuadratics will have a range above their vertex

142 xxy

)5,2( x

x ℝDomain Range 5y ,y ϵ ℝ

Domain Range

Domain & Range of Sqrt

3 xxf

xx ,3 ℝ 0xf ,f(x) ϵ ℝ

Can’t take sqrt of negative numbers (we want real numbers as answer)

0

3

Domain Range

Domain & Range of 1/x

xxf

1

xx ,0 ℝ 0xf ,f(x) ϵ ℝ

Match up Cards!

Some Things To Think On

• You have two minutes... What is the domain of possible x values for these functions...

35)(

)3(

1)(

xxf

xxxf

xєlR x≠0 and x≠ 3

xєlR x≥3/5

2 mins!!!1 min!!!

What Is The Domain and Range of… (3 minutes from my click)

3

1

542

xy

xxy Domain xЄR Min point (2, -9)Range yЄR and y ≥ -9

Domain xЄR x≠3Range yЄR and y≠0Output can never be 0(told you would need it later)

)4( xxy

xy sin1

)2(

1

xxy

xЄR0≤y≤2 (sinx between -1 and 1) yЄR

0≤x≤4 xЄR0≤y≤2 (roots of numbers from 0 to 4) yЄR

0<x<2 xЄRy>1 yЄR

Sketch & Find Domain & Range

Independent Study

• Exercise C page 13• Numerical Methods Test is next lesson Friday• Collecting Homework next lesson as well