• Functions and Their Graphs
Scratch Pad

A function is a rule that assigns to each element in a set A a single element in B, called f(x).

Example . f(x) = x ² .
This can sometimes be written f:x->x ² . In Maple this is written as

f:= x -> x^2

The domain of a function is the set of values for which f is defined, and the range of a function is the set of values that f takes on.

The graph of a function is the set of ordered pairs

{(x,f(x))|x in A}

The vertical line test consists of drawing a vertical line through a curve. The curve is a graph of a function if and only if every verticle line intersects the curve in at most one point.

Functions may be even or odd . A function is even if f(-x)=f(x) [symmetric]. A function is odd if f(-x)=-f(x) [anti-symmetric]

Functions may be combined by the operations of { + , - , x , and / }

The composition of two functions F(x) and G(x) is defined as FoG(x) = F(G(x))

• Types of Functions: Shifting and Scaling

The basic building blocks for functions that we consider are:

• constant
• power
• polynomial
• rational
• algebraic
• trigonometric
• exponential
• logarithmic
• transcendental

The fundamental graphical operations are shifting , stretching , and reflecting .

• Graphing Calculators and Computers
• Principles of Problem Solving

The fundamental principles of problem solving are

• Read the problem
• Identify relationships between the given information and the unknowns
• Try to recognize the familiar
• Try to recognize patterns
• Try to think of an analogous problem
• Introduce something extra
• Take special cases
• Work backwords
• Establish subgoals
• Use indirect reasoning
• Try mathematical induction

• A Preview of Calculus

Several motivating problems

• Area enclosed by a curve
• Tangent to a smooth curve
• Sum of a series (Xeno's paradox)

Last modified Mon Sep 2 20:24:58 CDT 1996