What is an even function?

An even function is a function that satisfies the property f(x) = f(-x) for all x in its domain. Geometrically, this means that the graph of an even function is symmetric with respect to the y-axis.

Here are some key characteristics of even functions:

• Symmetry: The graph of an even function is symmetric about the y-axis. This is the defining characteristic.

• Algebraic Test: To determine if a function is even, substitute -x for x in the function. If the resulting function is identical to the original function, then the function is even.

• Examples: Common examples of even functions include:

  • f(x) = x²
  • f(x) = cos(x)
  • f(x) = |x|
  • f(x) = x⁴ + 3x² + 1 (any polynomial with only even powers of x)

• Operations:

  • The sum, difference, and product of two even functions are even.
  • The composition of two even functions is even.
  • The derivative of an even function is an odd%20function.
  • The integral of an even function from -a to a is twice the integral from 0 to a: ∫^a_-a f(x) dx = 2∫^a₀ f(x) dx.

Understanding symmetry is crucial to grasping the concept of even functions. They are important in various areas of mathematics, including calculus and Fourier analysis. The concept is related to other types of function odd%20function.