Even Function:

A function $f(x)$ is even if, for all $x$ in the domain of $f$, the following condition holds: $f(-x) = f(x)$ This means that the graph of an even function is symmetric about the y-axis.

Examples of even functions:

• $f(x) = x^2$
• $f(x) = \cos(x)$

Odd Function:

A function $f(x)$ is odd if, for all $x$ in the domain of $f$, the following condition holds: $f(-x) = -f(x)$ This means that the graph of an odd function has rotational symmetry around the origin (180 degrees).

Examples of odd functions:

• $f(x) = x^3$
• $f(x) = \sin(x)$

Would you like me to explain more about how to determine if a function is even or odd? Let me know if you need any further details!

Related Questions:

1. How do you test if a function is even or odd algebraically?
2. What is the geometric interpretation of symmetry in even and odd functions?
3. Can a function be both even and odd? Why or why not?
4. What are some real-world examples of even and odd functions?
5. How do even and odd functions affect the integration of functions over symmetric intervals?

Tip:

A quick way to check if a function is even or odd is by plugging in values for $x$ and $-x$. For even, check if the function values are equal for $f(x)$ and $f(-x)$; for odd, check if they are negatives of each other.