Even functions
Even Functions
In mathematics, even functions are a special category of functions that exhibit symmetry with respect to the y-axis. This means that their graph is unchanged when reflected across the y-axis. The formal definition of an even function is based on the function's behavior with respect to positive and negative inputs.
Definition
A function ( f(x) ) is called an even function if for every ( x ) in the function's domain, the following condition holds:
[ f(-x) = f(x) ]
In other words, if you plug in the opposite of any number into an even function, you will get the same result as if you had plugged in the number itself.
Properties of Even Functions
- Symmetry: The graph of an even function is symmetric with respect to the y-axis.
- Origin: Even functions do not necessarily pass through the origin (0,0), unless the function also happens to be an odd function.
- Addition: The sum of two even functions is also an even function.
- Multiplication: The product of two even functions is an even function.
- Integration: The integral of an even function over a symmetric interval ([-a, a]) can be simplified to (2 \int_{0}^{a} f(x) \, dx).
Examples of Even Functions
Here are some common examples of even functions:
- ( f(x) = x^2 )
- ( f(x) = \cos(x) )
- ( f(x) = |x| ) (the absolute value function)
Table of Differences and Important Points
| Property | Even Functions | Odd Functions | General Functions |
|---|---|---|---|
| Definition | ( f(-x) = f(x) ) | ( f(-x) = -f(x) ) | No specific symmetry |
| Symmetry | Symmetric about the y-axis | Symmetric about the origin | No specific symmetry |
| Example | ( f(x) = x^2 ) | ( f(x) = x^3 ) | ( f(x) = x^2 + x ) |
| Integration over symmetric interval | ( \int_{-a}^{a} f(x) \, dx = 2 \int_{0}^{a} f(x) \, dx ) | ( \int_{-a}^{a} f(x) \, dx = 0 ) | No simplification |
Formulas Involving Even Functions
- Reflection Property: ( f(-x) = f(x) )
- Sum of Even Functions: If ( f(x) ) and ( g(x) ) are even, then ( h(x) = f(x) + g(x) ) is even.
- Product of Even Functions: If ( f(x) ) and ( g(x) ) are even, then ( h(x) = f(x) \cdot g(x) ) is even.
Examples to Explain Important Points
Example 1: Reflection Property
Consider the function ( f(x) = x^2 ). This function is even because:
[ f(-x) = (-x)^2 = x^2 = f(x) ]
The graph of ( f(x) = x^2 ) is a parabola that opens upwards and is symmetric about the y-axis.
Example 2: Sum of Even Functions
Let ( f(x) = x^2 ) and ( g(x) = \cos(x) ), both of which are even functions. Their sum ( h(x) = f(x) + g(x) = x^2 + \cos(x) ) is also an even function, as:
[ h(-x) = (-x)^2 + \cos(-x) = x^2 + \cos(x) = h(x) ]
Example 3: Integration of an Even Function
Consider the even function ( f(x) = x^2 ) and the symmetric interval ([-1, 1]). The integral of ( f(x) ) over this interval is:
[ \int_{-1}^{1} x^2 \, dx = 2 \int_{0}^{1} x^2 \, dx = 2 \left[ \frac{x^3}{3} \right]_0^1 = \frac{2}{3} ]
This demonstrates the simplification that occurs when integrating an even function over a symmetric interval.
Understanding even functions is crucial for analyzing the symmetry of graphs, simplifying integrals, and solving various mathematical problems. Remember that the key characteristic of an even function is its invariance when ( x ) is replaced with ( -x ).