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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.

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

Rapid Study Kit for "Title":
Flash Movie Flash Game Flash Card
Core Concept Tutorial Problem Solving Drill Review Cheat Sheet

"Title" Tutorial Summary :

This tutorial covers functions and their characteristics. The concept of graphing a function allows for a discussion of the inverse, composite and transformation of functions to be handled at a more manageable level.

A function is defined in its traditional way with the help of set notation and one to one correspondence. The domain of a function is needed to be discussed so the inverse of a function can be covered. Transformations of functions are introduced with the assistance of graphs and special notation.

Tutorial Features:

Specific Tutorial Features:

  • Graph of functions, ordered pair of functions, and functions are introduced with step by step examples.
  • 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:
Two sets and their correspondence
Functions and their definition
Domain of functions
One to one functions
Ordered pair of a function
Increasing and decreasing functions
Two dimensional linear functions
Definition of slope
Equation of a line
Parent Functions
Definition of parent functions
Transformations of functions
Vertical shifts
Horizontal shifts
Reflection in the y-axis
Reflection in the x-axis
Vertical stretching and shrinking
Horizontal stretching and shrinking
Composite and inverse functions

See all 24 lessons in college calculus, including concept tutorials, problem drills and cheat sheets:
Teach Yourself College Calculus Visually in 24 Hours