N9.+Use+function+notation,+domain,+and+range.

__Functions__
Functions are like machines that give out one //__**ONE**__// output. They are equations with only **__ONE__** solution, whereas general equations can have one of more solutions. The main thing to remember is that functions only have //__**ONE**__// solution For a given x, there's only **__ONE__** y. A good way to test if a graph is a function, do the Vertical Line Test. Take a straight object, hold it up and down (vertically) and bring it across your graph. If there are two points when your line crosses, then it's not a function.

__Domain and Range__
Domain is all possible x values Range is all possible y or f(x) values You can find these values by looking at tables, graphs or equations. Domain might look like 0 __>__ x __>__ 75 Range might look like -10 __>__ f(x) __>__ 104

__Examples:__

 * 1

Equation:

Graph:

Table:
 * **X** || **Y** ||
 * -2 || 4 ||
 * -1 || 1 ||
 * 0 || 0 ||
 * 1 || 1 ||
 * 2 || 4 ||

The __Domain__ for this problem is __>__ x __>__ The __Range__ for this problem is 0 __>__ y __>__

That's the answer for Domain because the graph, or x values, will always increase, they could go on for infinity. That's the answer for Range because you're never going to have a value below zero in your y column, but the numbers will always be increasing for infinity.


 * 2

Table: Assuming that this table doesn't continue on..
 * **X** || -2 || -1 || 0 || 1 || 2 || 3 || 4 || 5 || 6 ||
 * **f(x)** || 21 || 14 || 9 || 6 || 9 || 14 || 21 || 30 || 41 ||

1. f(6)=41 1a. This is the answer because when x equals 6, the value is 41. f(x)=? problems are asking you to find specific values for x. Therefore you find the find the x value on your table and the number next to it for your answer.

2. f(x)=14 then x=? x would equal 3 and -1. 2a. This is the answer because it's basically saying when y=14, what does x equal. So you find the f(x) value provided then find the x value that corresponds with it.

3. Domain= -2 __<__ x __<__ 6 3a. This is the answer because your x values aren't any lower then -2 or any higher than 6. Domain is basically finding the minimum and the maximum of a table.

4. Range= 6 __<__ f(x) __<__ 41 4a. This is the answer because your y values won't be any lower than 6 or any higher than 41. Range is almost the same thing as domain, except it's for your y values rather than your x values.

Is this an example of a function?
 * 3

No, it's not a function because for a given x there are more than one y values. The vertical line test would show the two spots where there are multiple y's for a given x.

The Mechanic Falls fil society is planning a special showing of a recently released film. The number of people who will attend the screening depends on the ticket price. Based on past experience, they decide to use the rule //N//(//p//)=400-20//p// to predict the number of people who will attend the screening if the ticket price is //p// dollars.
 * 5

A. What is the value of //N(//12) and what does it mean in this context?

N(12)=160 and this means that when the ticket price is $12, 160 people are expected to attend the showing.

This is the answer because you plug in 12 as your //p// value so the function would read N(12)=400-20(12) and you'd solve accordingly.

B. What value(s) of //p// satisfy N(p)=100 and what does it mean?

//p//= 15 and this means that if the ticket price was $15 then 100 people are expected to attend.

This is the answer because you plug 100 into the equation, making it 100=400-20//p//. After subtracting 400 from each side, you're left with -300=-20//p//. Next you divide each side by -20, to get //p//=15.

C. What would the domain and range be? Assuming that a negative amount of people will attend and you don't charge a negative amount for your tickets.

Domain: 0 __<__ //p// __<__ 20 Range: 0 __<__ N(//p//) __<__ 400

You get this answer by using your equation. The x, value won't be more than 20 or less than zero and the y won't be more than 400 or less than 0. You get these values from the equation. The x value is what's attached to the x, or in this case "p". The y value is what's by itself, generally next to the x value.

Common Mistakes:

 * When there isn't a graph or table, pulling information from equations can often lead to confusion
 * Infinities are hard to understand, it's tricky to figure out if things go on forever or decide when they will stop
 * It's normal to assume things, like Domain and Range are infinite, when they really aren't
 * When you're given the equation and the domain, it's difficult to find the range
 * It's hard to figure out the difference between f(x)=7 and f(4)=?
 * Deciding if somethings a function or not

For More Information...
http://www.purplemath.com/modules/fcns2.htm pg 345 #1-5 for Domain and Range practice pg 346 #4 for Function Notation practice