Even and Odd Functions

Even Functions: An even function is one where the output value remains the same even when the input is negated. Replacing x with −x in the function, the result will not change. This shows that the function has symmetry about the y-axis. Functions like f(x) =x2  yields the same result, for positive or negative inputs, e.g., f(2) = f(–2) = 4.

Odd Functions: These functions exhibit symmetry about the origin. For example, an odd function satisfies the condition f(−x) = −f(x) for all x in its domain, indicating rotational symmetry about the origin. 
 

To determine if a function is even or odd, follow these steps: Even Function: If f(−x) = f(x) for all x in its domain, the function is even. Odd Function: If f(−x) = −f(x) for all x in its domain, the function is odd.

Even Function
Definition: A function f(x)  is even if f(−x) = f(x) for all x in its domain.

Graphical Symmetry: Graph of an even function is symmetric about the y-axis
Examples: f(x) = x2 , f(x) = cos⁡( x), f(x) = ∣x∣

Odd Function
Definition: A function f(x) is odd if f(−x) = −f(x)  for all x in its domain.

Graphical Symmetry: The graph of an odd function has rotational symmetry about the origin.
Examples: f(x) =x3, f(x) = sin (x), f(x) = x

Both Even and Odd
Zero Function: The only function that is both even and odd is f(x) = 0 since f(−x) = f(x) = −f(x) = 0

Neither Even Nor Odd
Example: f(x) =(-x)3+(-x)+1
f(−x) = −x3−x+1

Neither f(−x) = f(x) nor f(−x) = −f(x), so it's neither even nor odd.
 

Trigonometric functions, such as sine, cosine, tangent, cotangent, secant, and cosecant, can be categorized based on their symmetry and behavior. In even functions, the condition f(−x) = f(x) means the graph is symmetric about the y-axis. In odd functions, the condition f(−x) = −f(x) , is such that the graph has rotational symmetry about the origin.

Even Functions
A function f(x) is defined as even if it satisfies the condition:
f(−x) = f(x)
This tells us that the graph of the function is symmetric about the y-axis
Trigonometric Examples:

• Cosine Function: cos⁡(−x) = cos⁡(x)
• Secant Function: sec⁡(−x) = sec⁡(x)

These functions exhibit symmetry about the y-axis; their values remain unchanged when the input angle is negated.

Odd Functions
A function f(x) is defined as odd if it satisfies the condition:
f(−x) = −f(x)
This shows that the graph of the function has origin symmetry.

Trigonometric Examples:

• Sine Function: sin⁡(−x) = −sin⁡(x)
• Tangent Function: tan⁡(−x) = −tan⁡(x)
• Cosecant Function: csc⁡(−x) = −csc⁡(x)
• Cotangent Function: cot⁡(−x) = −cot⁡(x)

Even and odd functions exhibit defined symmetry properties that clarify the evaluation of integrals.
 

Even Functions
For a continuous even function f(x) satisfying f(−x) = f(x), the integral above the symmetric interval [−a, a] can be simplified:
-aa  f(x) dx=2 0a f(x) dx
This symmetry simplifies calculations by doubling the integral from 0 to a, as the function is identical on both sides of the y-axis.
 

Odd Functions
For a continuous odd function f(x) satisfying f(−x) = −f(x), the integral over the symmetric interval [−a, a] equals zero:
-aa  f(x) dx=0

This is because the areas above and below the x-axis cancel each other out due to symmetry.

Addition & Subtraction

The sum of two even functions is even, as it retains symmetry about the y-axis.
            Even + Even = Even

If you add two functions that have rotational symmetry about the origin, the result will also have rotational symmetry about the origin.

           Odd + Odd = Odd

When you combine a function with y-axis symmetry and one with origin symmetry, it doesn't result in a function that is symmetric about the y-axis or origin.

            Even + Odd = Neither

Multiplication & Division
Multiplying two functions that are symmetric about the y-axis results in a function that is also symmetric about the y-axis.
Even × Even = Even

Multiplying two functions with rotational symmetry about the origin results in a function that is symmetric about the y-axis.
Odd × Odd = Even

Multiplying a function symmetric about the y-axis with one having rotational symmetry about the origin results in a function with rotational symmetry about the origin.
Even × Odd = Odd

Dividing two such functions does not necessarily preserve this symmetry.
Even ÷ Even = Even

Multiplying functions symmetric about the y-axis keep symmetry, but dividing them doesn't necessarily do so if the quotient is undefined at certain points.
Odd ÷ Odd = Even

Dividing a function symmetric about the y-axis by one with rotational symmetry about the origin results in a function with rotational symmetry about the origin.
Even ÷ Odd = Odd

Composition 
Composing two functions which are symmetric about the y-axis results in a function symmetric.
Even ∘ Even = Even

Composing two functions with rotational symmetry about the origin results in a function which has rotational symmetry about the origin.
Odd ∘ Odd = Odd

Composing a function symmetric about the y-axis with one having rotational symmetry about the origin results in a function which is symmetric about the y-axis.
Even ∘ Odd = Even
 

An even function exhibits symmetry about the y-axis, its graph remains unchanged when reflected across the y-axis, and the function's values are identical for every pair of opposite x-values.

Even Functions Graph
 

An even function is a type of mathematical function that behaves symmetrically around the y-axis. This means, reflecting its graph over the y-axis, the shape would remain unchanged. For every point (x, y) on the graph of an even function, the point (–x, y) is also on the graph, mirroring across the y-axis.
 

An odd function is a type of mathematical function that has a specific kind of symmetry. This symmetry means that if you rotate the graph of the function 180 degrees around the origin, the graph will look the same.
 

Even and odd functions are core concepts in mathematics, reflecting symmetry in various real-world situations. Understanding the real-life applications of even and odd functions have applications in patterns in nature, technology, and everyday life.

• Electromagnetic Theory:  In electromagnetic theory, odd functions model current distributions with rotational symmetry, such as f(x) = x for magnetic field behavior.

• Symmetrical Sound Waves: When certain musical instruments produce sound, the waveforms can be symmetric. This symmetry in sound waves is analogous to even functions, where the pattern remains unchanged when reflected across the y-axis.

• Optical Lenses: The design of some optical lenses ensures that light passing through them is symmetrically refracted. This symmetrical behavior is related to the properties of even functions.

• Torque and Angular Displacement: The relationship between the torque applied to a wheel and its angular displacement is an example of an odd function: applying torque in one direction results in angular displacement in the same direction, and vice versa, satisfying f(−x) = −f(x).

• Vibrating Guitar String: When a guitar string vibrates, the displacement of the string can be described by odd functions, where the displacement at one point is the negative of the displacement at the opposite point.