Pre Calculus Functions and Graphs

Functions A function is a relation where each element of the domain is paired with exactly one element of the range independent variable - x dependent variable - y domain - set of all values taken by independent variable range - set of all values taken by the dependent variable

Mapping

Representing Functions notation - f(x) numerical model - table/list of ordered pairs, matching input (x) with output (y) US Prison Polulation (thousands) YearTotalMaleFemale

graphical model - points on a graph; input (x) on horizontal axis … output (y) on vertical algebraic model - an equation in two variables

Vertical Line Test

Finding the range implied domain - set of all real numbers for which expression is defined example: Find the range

Continuity function is continuous if you can trace it with your pencil and not lift the pencil off the paper

Discontinuities point discontinuity –graph has a “hole” –called removable –example

jump discontinuity - gap between functions is a piecewise function example

infinite discontinuity - there is a vertical asymptote somewhere on the graph example

Finding discontinuities factor; find where function undefined sub. each value back into original f(x) results …

Increasing - Decreasing Functions function increasing on interval if, for any two points decreasing on interval if constant on interval if

Example:

Boundedness of a Function

Extremes of a Function local maximum - of a function is a value f(c) that is greater than all y- values on some interval containing point c. If f(c) is greater than all range values, then f(c) is called the absolute maximum

local minimum - of a function is a value f(c) that is less than all y-values on some interval containing point c. If f(c) is less than all range values, then f(c) is called the absolute minimum

A B C D E F G H I J K local maxima Absolute maximum Absolute minimum local minima

Example: Identify whether the function has any local maxima or minima

Symmetry graph looks same to left and right of some dividing line can be shown graphically, numerically, and algebraically graph: x f(x) numerically

algebraically even function –symmetric about the y-axix –example

odd function –symmetric about the origin –example

Additional examples: even / odd

Asymptotes horizontal - any horizontal line the graph gets closer and closer to but not touch vertical - any vertical line(s) the graph gets closer and closer to but not touch Find vertical asymptote by setting denominator equal to zero and solving

End Behavior A function will ultimately behave as follows: –polynomial … term with the highest degree –rational function … f(x)/g(x) take highest degree in num. and highest degree in denom. and reduce those terms –example