What is a function?

A function is a relation (a set of ordered pairs) where every input (x) has only one output (y).

To determine functionality, use the vertical line test. If the line crosses more than once, then the line is not a function.

*FindingDomain and Range**:*

Domain: X's that work in a function

Range: Y's that work in a function

If the point on the graph is filled in, that the function includes that number. If the point is not filled in, then the function does not include that number.

Increasing, Decreasing, and Constant- ALWAYS use parentheses because it is a line and it always includes whatever point it is on.

Example of decreasing infinity and negative infinity (where decreasing describes the y value and infinity describes the x value)

A function is increasing if the y value is getting bigger from left to right.

A function is decreasing if the y value is getting smaller from left to right.

A function is constant if the y value stays the same.

Relative Maximum and Minimum:

Plug in function in your calculator

1. The "Y=" button should be on the top left-hand side of the calculator. Press that and enter your function in terms of y (so that y is only on one side with no other nomials or coefficients other than 1).

2. To find a relative maximum or minimum, press "2nd" then "CALC." Scroll down to minimum or maximum, depending on what you're looking for.

3. Press enter, then, using the left/right arrows, move the blinking dot so that it is left of the relative maximum/minimum. Press enter, then repeat the process for the rigt side.

4. Press enter twice and the ordered pair of the relative minimum or maximum will be at the bottom of the screen.

Maximum- the point on the graph where the y value is highest (not including infinity).

Minimum- the point on the graph where the y value is lowest (not including negative infinity).

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**How to determine if a function is even, odd, or neither.**

There are __two ways__ to determine if a function is even, odd, or neither: by looking at it's graph and by looking at it's equation.

__Graph:__

If the graph of the function is symmetrical over the y-axis, it is even.

If the graph of the function is symmetric about the origin, it is odd. That is, if the graph can be rotated about the origin 180 degrees and lies on itself.

__Equation:__

If all the exponents of the variable are even, the function is even.

--f(x) = 5x^{4} + x^{2} + 4

Note: This works as an even function because the "4" has an x with an exponent of 0, since everything raised to the 0th power equals 1. Therefore, all the exponents are even.

If all the exponents of the variable are odd, the function is odd.

--f(x) = 6x^{5} + 2x^{3} + x

Note: The last variable x has an exponent of 1, since everything raised to the first power is itself. Therefore, all the exponents are odd.

**Even**

**Odd**

Links to further help Graphing of Functions:

http://www.uncwil.edu/courses/mat111hb/functions/graphs/graphs.html

http://www.purplemath.com/modules/fcns2.htm

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