INDEX
Numbers and Sets
Complex Numbers
Exponentiation
Algebraic Transformations
Algebraic Equations and Inequalities
Discrete Algebra
Basic Formulas
Graphics of Basic Functions
Functions
Ship 2

Basic Conceptions

Cartesian Coordinate System

Domain and Range

Inverse Functions

Even-Odd Symmetry of Functions

Periodicity of Functions

Exponential and Logarithmic Functions

Exponential Functions

Logarithmic Functions
  Natural Logarithms

Sets and Intervals

Hyperbolic Functions

Even-Odd Symmetry of Functions
Key Topics Remaining: » Exponential Functions » Logarithmic Functions » Natural Logarithms

A function  f ( x )  is called an even function if

f ( – x ) = f ( x )

for any  x  in the domain of  f ( x ).

Examples of even functions:

x2,    x4,    x6,    1 / x2 ,    | x |

cos x,    cosh x,    cos2 x,    cosh2 x.

Graphics of even functions are symmetric with respect to the  y-axis.

A function  f ( x )  is called an odd function if

f ( – x ) = – f ( x )

for any  x  in the domain of  f ( x ).

Examples of odd functions:

x,    x3,    x5,    1 / x ,    1 / x3 ,    ,    ,

sin x,    tan x,    cot x,    tanh x,    coth x.

Graphics of odd functions are symmetric with respect to the origin.

Even-odd properties.
  1. The sum and product of any number of even functions is an even function.

  2. The sum of odd functions is an odd function.

  3. The product of two odd functions is an even function.

  4. The product of an even function and an odd function is an odd function.
However, the most part of functions are neither even nor odd.
Examples
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