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

A set of ordered pairs. (x, y) What is a relation? A set of ordered pairs. (x, y) Could you represent a relation another way?

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. Notice anything about the order of the domain/range?

Functions Discrete- a graph which consists of points which are not connected.

A function that is traceable! Continuous Data What do you think of when you hear the word continuous? A function that is traceable! Examples: Lines…Parabolas…any others?

Discontinuous Data What 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!

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?? Do the x-values repeat? 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 complete sentence. 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: {(1,2),(2,4),(3,5)(0,5)} {(0,4),(2,4),(1,3),(2,5)} {(&,*),($,%),(#,^),(@,*),(#,@)} Yes No

Examples State whether each relation or graph below is a function: no yes

Examples State whether each relation or graph below is a function: (5,5) is open yes no

(a, b) a< x <b Open Interval Parenthesis-NOT inclusive An open interval is the set of all real numbers that lie strictly between two fixed numbers a and b. Parenthesis-NOT inclusive is always open (a, b) a< x <b ***think about open dots on a number line!

[a, b] a< x <b Closed Interval Brackets mean-inclusive 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 <b Brackets mean-inclusive **think about closed dots on a number line!

(a, b] [a, b) a< x <b a< x <b 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 -1< x <4 x< 1 or x > 5 Fill in the missing parts in the chart below. Inequality Interval Notation Graph -1< x <4 x< 1 or x > 5

Graph Example 2 Determine the domain, range, and continuity of the graph below.

Graph Example 3 Determine the domain, range, and continuity of the graph below.

Graph Example 4 Determine the domain, range, and continuity of the graph below.

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.

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?)

Increasing/Decreasing Example 6 b. Determine the interval(s) of x in which f(x) in decreasing. (between what two x values is the function decreasing?)

Increasing/Decreasing Example 6 c. Determine the interval(s) of x in which f(x) is positive. (between what two x values are the y-values positive?)

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