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Functions and Graphs

 Topic Review on "Title": Function (from a set X to a set Y): A correspondence that associates with each element x (independent variable) of X and a unique element y (dependent variable) of Y. Notation: or . Domain of a function: The data set of all real numbers for which the correspondence makes sense. One to one functions: is a function from to . Increasing functions: If S is a subset of X and whenever, in S, then is an increasing function in S. Decreasing functions: If S is subset of X and whenever, in S, then is a decreasing function in S. Slope: If  is a line which is not parallel to the -axis and if and are distinct points on, then the slope of  is given by: .  Equation of a curve: Suppose of a curve composed of points whose coordinates are  for 1, 2,…. If there is an equation, by which all the can be calculated through substituting, the equation is called the equation of the curve. Equation of a line: If the slope of a line is given (denoted by ) and if a point on the line is given (coordinate is ) , the line equation would be . Vertical shifts: If is a real number, the graph of is the graph of shifted upward units for or shifted downward for . Horizontal shifts: If is a real number, the graph of is the graph of shifted to the right units for or shifted to the left units for . Reflection in the y-axis: The graph of the function is the graph of reflected in the y-axis. Reflection in the x-axis: The graph of the function is the graph of reflected in the x-axis. Vertical stretching and shrinking: If is a real number, the graph of is the graph of stretched vertically by for or shrunk vertically by for . Horizontal stretching and shrinking: If is a real number, the graph of is the graph of stretched horizontally by for or shrunk horizontally by for . Composite functions: If is a function from to and is the function from to , then the composite function is the function from to defined by . Inverse functions: Let be a one to one function from to . Then, a function  from to is called the inverse function of if for all in and for all in

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 "Title" Tutorial Summary : This tutorial describes functions and their properties through the use of their properties and relations. Graphing a function makes the development of composite, transformation and inverse functions easier rather than difficult. The definition of a function in set notation form is formulated with the use of the correspondence between two data sets. The inverse of a function is discussed with the use of a domain of a function. The idea behind how a function increases or decreases is discussed with the help of graphs and examples.

 Tutorial Features: Specific Tutorial Features: • Step by step examples are shown to introduce functions, ordered pair of functions, and graphs of functions. • Sketches of functions with visual aids are shown in the tutorial to help introduce concepts such as decreasing functions. Series Features: • Concept map showing inter-connections of new concepts in this tutorial and those previously introduced. • Definition slides introduce terms as they are needed. • Visual representation of concepts • Animated examples—worked out step by step • A concise summary is given at the conclusion of the tutorial.

 "Title" Topic List: Functions Correspondence between two sets Definition of functions Domain of functions One to one functions Ordered pair of a functionIncreasing and decreasing functionsLinear functions in two variables Definition of slope Equation of a lineParent Functions Definition of parent functionsTransformations of functions Vertical shifts Horizontal shifts Reflection in the y-axis Reflection in the x-axis Vertical stretching and shrinking Horizontal stretching and shrinkingComposite and inverse functions

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