Tools to use Calculus
Functions		Understanding = Effort x ( clarity / time )
Calculus is about change. Eventually, we will talk about zero and infinity. At these static points change is still defined. The great philosopher, Anonymous, once said, "The only Constant in the Universe is Change". To be as vague as possible we can say something happens to something and we perceive a change. To understand "something" we will take a stand and define Function. The definition focuses on the change that a function causes to produce an output from an input. Each element of output is individually targeted and hit by a specific element of input. A function is unable to direct an input value to more than one value of output. Only valid input values are used by a function. The set of valid input values is called the Domain and the set of output values produced is called the Target or Range. Thus a Function is a rule of correspondence between two sets such that there is a unique element in the Target set assigned to each element of the Domain set. The realm of a function is not simply limited to its process, it also dictates which values it will use as input to produce output. This is why the definition stated in the last sentence of the paragraph above starts in the middle, goes to the end and finally comes back to the beginning.		"Figuring out the properties of a function when given its input and output values is like building a jigsaw puzzle. First you sort out the pieces, then see how they fit together. Only thing is; the pieces may be a little out of focus to begin with."
The Domain and Range own their elements in the manner specified by their Interval. For now we will consider Real-Value Functions such that the Domain is an element of the Real Numbers, D  , and so is the Range, T  . The function dictates which Real values are used for D and T. The Domain and Range are subsets of the Reals, D and T .
The Interval defines how close to a value in that D and T will be. If they own the end-point, the Interval is closed.		The numbers we take for granted are subsets. N Z Q R C N is the set of Whole Numbers {0, 1, 2, 3, ...} Z is the set of Integers {..., -2, -1, 0, 1, 2, ...}
If they do not own the end-point, the Interval is opened.		Q is the set of Rational Numbers commonly called fractions of the form p/q, where p and q are Integers and q does not equal zero.
The Interval can also be half-open or half-closed. The choice of phrasing is not a reflection of optimism or pessimism it emphasizes what is of interest at the moment. An infinite interval replaces the dots or circles with an arrow which of course can also be half-open or half-closed.		R is the set of Real Numbers represented by infinite decimals, not necessarily repeating, such as 2, or e. C is the set of Complex Numbers which have the form a + ib, where i = -1.
Symmetry: Even, Odd or neither
A function has a Domain and a Range. When a function is mapped in the Cartesian Coordinate Plane it inherits some form of symmetry. Recall,
, defines three aspects of symmetry. Functions that satisfy the condition f(-x) = f(x) are symmetric about the y-axis and are called even functions. Those that that hold true when f(-x) = -f(x) are symmetric about the origin and are called odd functions. Graphs of functions that contain only even powers of x will be even functions, however there is not a general rule concerning only odd powers of x corresponding to odd functions.
Functions can be grouped into families. Each member of a family produces a similar graph when plotted on the Cartesian Plane. By moving, stretching and rotating a function we can model a vast array of phenomena. It is beneficial to gain a little experience with manipulating some simple functions before we look at families of functions.
Circles and Parabolas - section four		Tools to use Calculus - section topics