Even and Odd 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 \(f(x) =x^2\)  yields the same result, for positive or negative inputs, e.g., \(f(2) = f(–2) = 4\).

Odd 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: \(f(x) = x^2\), \(f(x) = cos(x), f(x) = |x|\)

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: \(f(x) = x^3, f(x) = sin (x), f(x) = x\)

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: \(f(x) = (-x)^3 + (-x) + 1\\ f(−x) = −x^3−x+1 \)

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. Here are a few properties of even and odd function: 
 

• 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: 
  
  \(\int_{-a}^{a} f(x) \,dx = 2 \int_{0}^{a} f(x) \,dx\)
  
  ​​​​​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: 
   \(\int_{-a}^{a} f(x) \,dx = 0\)
  
  This is because the areas above and below the x-axis cancel each other out due to symmetry.

Even functions are symmetric about the y-axis, and odd functions are symmetric about the origin. These properties help simplify graphs and calculations.

 

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.
  
  \(\text{Even} \circ \text{Even} = \text{Even}\)

• Composing two functions with rotational symmetry about the origin results in a function which has rotational symmetry about the origin.
  
  \(\text{Odd} \circ \text{Odd} = \text{Odd}\)

• 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.
   

   
  \(\text{Even} \circ \text{Odd} = \text{Even}\)

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.
 

Odd Function Graph
 

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 important concepts in mathematics that help us understand symmetry in graphs and patterns in algebraic expressions. Recognizing whether a function is even, odd, or neither can make solving equations and analyzing graphs much faster. Here are some tips and tricks for students:

• Even functions are symmetric about the y-axis. Imagine folding the graph along the y-axis both halves should match perfectly. Odd functions are symmetric about the origin. If you rotate the graph 180° around the origin, it should look the same.
  
   
• To check if a function is even or odd, replace x with -x. If \(f(-x) = f(x)\), it is even. If \(f(-x) = -f(x)\), it is odd. 
  
   
• There are certain functions which are commonly even or odd. Powers of x like \(x^2, x^4\)are even, while powers like \(x, x^3\) are odd. 
  
   
• When adding or multiplying functions, use these rules:
  
  \(Even ± Even = Even \\ \\ \ \\ Odd ± Odd = Odd\\ \\ \ \\ Even × Even = Even\\ \\ \ \\ Odd × Odd = Odd\\ \\ \ \\ Even × Odd = Odd\\\)
   
• Always remember that the zero function \(f(x) = 0\) is both even and odd. 

Even and odd functions are important in mathematics and appear in many real-world situations, showing symmetry in nature, technology, and daily life. Understanding the real-life applications of even and odd functions have applications in patterns in nature, technology, and everyday life.

• Electromagnetic Theory:  : Odd functions model current distributions with rotational symmetry, such as f(x) = x, which describes certain magnetic field behaviors.

• Symmetrical Sound Waves: Some musical instruments produce sound waves that are symmetric, similar to even functions, where the waveform remains unchanged when reflected across the y-axis.

• Optical Lenses: Certain lenses are designed so that light passing through is symmetrically refracted, reflecting the properties of even functions.

• Torque and Angular Displacement: The relationship between applied torque and angular displacement often follows an odd function pattern: torque in one direction causes displacement in the same direction, satisfying \(f(−x) = −f(x)\).

• Vibrating Guitar String: The displacement of a vibrating guitar string can be represented by odd functions, where the movement at one point is the negative of the movement at the opposite point.