1. What is Function Symmetry?

Function symmetry refers to whether a function is even, odd, or neither. An even function satisfies f(-x) = f(x), while an odd function satisfies f(-x) = -f(x). Symmetry helps in analyzing and graphing functions.

2. How Does the Calculator Work?

The calculator checks function symmetry by evaluating:

Where:

• \( f(x) \) — The input function
• \( x \) — The variable in the function

Explanation: The calculator substitutes -x for x and compares the result to the original function.

3. Importance of Function Symmetry

Details: Knowing if a function is even or odd simplifies integration, Fourier analysis, and helps predict function behavior. Even functions are symmetric about the y-axis, while odd functions have rotational symmetry.

4. Using the Calculator

Tips: Enter the function using standard mathematical notation (e.g., "x^2" or "sin(x)"). Specify the variable (default is x). The calculator will determine if the function is even, odd, or neither.

5. Frequently Asked Questions (FAQ)

Q1: Can a function be both even and odd?
A: Yes, only the zero function (f(x) = 0) is both even and odd.

Q2: What are examples of even functions?
A: x², cos(x), and |x| are common even functions.

Q3: What are examples of odd functions?
A: x³, sin(x), and x are common odd functions.

Q4: How is symmetry useful in integration?
A: For even functions, ∫[-a,a]f(x)dx = 2∫[0,a]f(x)dx. For odd functions, ∫[-a,a]f(x)dx = 0.

Q5: Can non-polynomial functions be even or odd?
A: Yes, trigonometric and other transcendental functions can also exhibit symmetry.