F UNCTIONS D EFINED Come and Get it!

Definition of Function Function is a relation in which each element of the domain is paired with exactly one element of the range. The domain is the set of all first elements of ordered pairs ( x -coordinates). The range is the set of all second elements of ordered pairs ( y -coordinates). Function notation is the way a function is written. It is meant to be a precise way of giving information about the function without a rather lengthy written explanation. The most popular function notation is f ( x ) which is read " f of x ".

A symbol that stands for an arbitrary input is called an independent variable, while a symbol that stands for an arbitrary output is called a dependent variable. The most common symbol for the input is x, and the most common symbol for the output is y.

An open interval does not include its endpoints, and is indicated with parentheses. For example (0,1) means greater than 0 and less than 1. A closed interval includes its endpoints, and is denoted with square brackets. For example [0,1] means greater than or equal to 0 and less than or equal to 1.

Vertical Line Test A test use to determine if a relation is a function. A relation is a function if there are no vertical lines that intersect the graph at more than one point.relationvertical linesgraph point Horizontal Line Test A test use to determine if a function is one-to-one. If a horizontal line intersects a function's graph more than once, then the function is not one-to- one.functionone-to-onehorizontallinegraph

Extremum An extremum is a maximum or minimum. An extremum may be local (a.k.a. a relative extremum; an extremum in a given region which is not the overall maximum or minimum) or global. Functions with many extrema can be very difficult to graph.maximumminimumlocalmaximumminimum globalgraph

Even Functions A function is "even" when: f(x) = f(−x) for all x In other words there is symmetry about the y- axis (like a reflection):symmetry about the y- axis Odd Functions A function is "odd" when: −f(x) = f(−x) for all x Note the minus in front of f: −f(x). And we get origin symmetry:origin symmetry

The term "composition of functions" (or "composite function") refers to the combining of functions in a manner where the output from one function becomes the input for the next function. In math terms, the range (the y -value answers) of one function becomes the domain (the x -values) of the next function. The notation used for composition is: f(g(x) and is read " f composed with g of x " or " f of g of x ". Notice how the letters stay in the same order in each expression for the composition. f ( g ( x )) clearly tells you to start with function g (innermost

In mathematics, an inverse function is a function that "reverses" another function. That is, if f is a function mapping x to y, then the inverse function of f maps y back to x. In mathematics, a piecewise - defined function (also called a piecewise function or a hybrid function ) is a function which is defined by multiple sub functions, each sub function applying to a certain interval of the main function's domain (a sub-domain).

Continuous Function – For all defined domain values there exists a range value. Otherwise a function is said to discontinuous.