To determine whether a function is even, odd, or neither, you can follow these steps:

1. Even Function Test:

A function $f(x)$ is even if it satisfies the condition: $f(-x) = f(x)$ This means that the function's graph is symmetric about the y-axis.

2. Odd Function Test:

A function $f(x)$ is odd if it satisfies the condition: $f(-x) = -f(x)$ This means that the function's graph is symmetric about the origin.

Steps to check:

• Step 1: Substitute $-x$ into the function, i.e., find $f(-x)$.
• Step 2: Compare $f(-x)$ with $f(x)$ and $-f(x)$.
  • If $f(-x) = f(x)$, the function is even.
  • If $f(-x) = -f(x)$, the function is odd.
  • If neither condition holds, the function is neither even nor odd.

Example 1: $f(x) = x^2$

• $f(-x) = (-x)^2 = x^2$
• Since $f(-x) = f(x)$, the function is even.

Example 2: $f(x) = x^3$

• $f(-x) = (-x)^3 = -x^3$
• Since $f(-x) = -f(x)$, the function is odd.

Example 3: $f(x) = x^2 + x$

• $f(-x) = (-x)^2 + (-x) = x^2 - x$
• Since $f(-x) \neq f(x)$ and $f(-x) \neq -f(x)$, the function is neither even nor odd.

Would you like more details on this topic?

Here are 5 related questions for deeper understanding:

1. What are examples of neither even nor odd functions?
2. How do you determine if a piecewise function is even or odd?
3. Can a function be both even and odd?
4. How does symmetry in graphs help determine if a function is even or odd?
5. What are the practical applications of even and odd functions in real-world problems?

Tip: When checking for symmetry, sketching the graph of the function can help visually confirm whether it is even, odd, or neither.