Thinker….

Lew is playing darts on a star-shaped dartboard

in which two equilateral triangles

trisect the sides of each other as shown. Assuming that a dart

hits the board, what is the probability that it

will land inside the hexagon?

Solution to the “Thinker”

1/2. As shown, each triangle can

be reflected to the interior of the hexagon in such a way that the triangle areas are equal to the area of the hexagon. In this manner, the area of the hexagon is half that of the entire dartboard.

Functions

Domain & Range

Increasing & Decreasing

What is a relation?

A set of ordered pairs. (x, y)

Could you represent a relation another way?

)}1,3(),1,0(),2,2(),2,2{(

Domain vs. Range

• Domain- the set of x-coordinates of a relation or function.

• Range- the set of y-coordinates of a relation or function.

}3,0,2{: D

}2,1,2{: RNotice anything about the order of the domain/range?

Functions• Discrete- a graph which consists of

points which are not connected.

Continuous DataWhat do you think of when you hear the word continuous?

A function that is traceable! Examples: Lines…

Parabolas…any others?

Discontinuous DataWhat do you think of when you hear the word discontinuous?

A function that is not traceable. (must pick up your pencil)

Would discrete data be continuous or discontinuous?

Piecewise Functions-a graph which consists of line segments or pieces of other nonlinear graphs.

Would piecewise functions be continuous or discontinuous?

Would piecewise functions be discrete?

What is a function?A function is a relation in which each element of the domain is paired with exactly one element from the range (no duplicate x-values). A function is denoted as f(x) or pronounced “f of x”.

Some relations are functions…some are not…

But…how do we know? Let’s find out!

)}8,4(),3,3(),0,3(),1,9{( f(x)=x

-3 -2 -1 1 2

-3

-2

-1

1

2

x

y

f(x)=-x̂ 2 + 5

-8 -6 -4 -2 2 4 6 8

-6

-4

-2

2

4

6

x

y

f(x)=sin(x*10.5/10)

-3 -2 -1 1 2 3

-2

-1

1

2

x

y

Draw these two graphs.

• Vertical Line Test!• Touches it once, it IS a function!• Touches >1, it is NOT a function!

What if it’s not a graph??

)}8,4(),3,3(),0,3(),1,9{(

State weather the following relation is a function or not.

•Do the x-values repeat?

•No they don’t….YES it is a function!

•Yes they repeat…NO it’s not a function!

That’s too hard to remember, can I just graph the points and use the VLT?

Example 1

• Determine whether the relation {(-1, ), (-1, ), (0, 1)} is a function. • Justify your decision in a completesentence.

This relation is not a function since two values of -1 will not pass the vertical line test.

Examples

• State whether each relation or graph below is a function:

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

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

3. {(&,*),($,%),(#,^),(@,*),(#,@)}

Yes

No

No

Examples•State whether each relation or graph below is a function:

no yes no

Examples•State whether each relation or graph below is a function:

(5,5) is open

yes no

Open Interval

• An open interval is the set of all real numbers that lie strictly between two fixed numbers a and b.

(a, b)

a< x <b

Parenthesis-NOT inclusive

is always open

***think about open dots on a number line!

Closed Interval• A closed interval is the set of all

real numbers that lie in between and contain both endpoints a and b.

[a, b]

a< x <bBrackets mean-inclusive

**think about closed dots on a number line!

Half-Open Intervals• Half-open intervals are intervals

that contain one but not both endpoints a and b.

[a, b)

a< x <b

(a, b]

a< x <b

What do we use this for?

• **Intervals will be used to define the domain and range of given functions or graphs which are continuous and/or increasing and decreasing intervals.

OTHER SYMBOLS TO KNOW:

U : union symbol used to join more than one interval together.

0 (zero): neither positive nor negation.

Number Line Examples

InequalityInterval Notation Graph

-1< x <4

x< 1 or x > 5

Fill in the missing parts in the chart below.

Graph Example 2

Determine the domain, range, and continuity of the graph below.

x

y

Graph Example 3

Determine the domain, range, and continuity of the graph below.

x

y

Graph Example 4

Determine the domain, range, and continuity of the graph below.

x

y

Graph Example 5

Determine the domain, range, and continuity of the graph below.

Increasing/Decreasing• Increasing intervals occur when. . .reading

a graph left to right, the interval in which the function is rising.

Decreasing intervals occur when. . .reading a graph left to right, the interval in which the function is falling.

x

y

x

y

Increasing/Decreasing Example 6

a. Determine the interval(s) of x in which f(x) in increasing. (between what two x values is the function increasing?)

x

y

Increasing/Decreasing Example 6

x

yb. Determine the interval(s) of x in which f(x) in decreasing. (between what two x values is the function decreasing?)

c. Determine the interval(s) of x in which f(x) is positive. (between what two x values are the y-values positive?)

Increasing/Decreasing Example 6

x

y

Reflection

•Write a question in your notebook about something Mrs. Gromesh taught today that you aren’t 100% on understanding yet.