These are two questions in which we need to Determine whether function is even or odd.

To determine whether a function is even, odd, or neither, we can use specific mathematical definitions and properties. Let's break this down step by step.

Understanding Even and Odd Functions

A function \( f(x) \) is classified based on its symmetry properties:

• Even Function: A function is even if it satisfies the condition \( f(-x) = f(x) \) for all \( x \) in its domain. This means that the graph of the function is symmetric about the y-axis.
• Odd Function: A function is odd if it meets the condition \( f(-x) = -f(x) \) for all \( x \) in its domain. This indicates that the graph of the function is symmetric about the origin.

Steps to Determine the Nature of a Function

To analyze whether a given function is even, odd, or neither, follow these steps:

1. Substitute -x into the function: Replace every instance of \( x \) in the function with \( -x \).
2. Simplify the expression: After substitution, simplify the resulting expression.
3. Compare results:
   • 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 Analysis

Let’s consider two functions to illustrate this process:

Example 1: \( f(x) = x^2 + 3 \)

1. Substitute -x:

\( f(-x) = (-x)^2 + 3 = x^2 + 3 \)

2. Compare:

Since \( f(-x) = f(x) \), this function is even.

Example 2: \( g(x) = x^3 - x \)

1. Substitute -x:

\( g(-x) = (-x)^3 - (-x) = -x^3 + x \)

2. Compare:

Now, check if \( g(-x) = -g(x) \):

\( -g(x) = - (x^3 - x) = -x^3 + x \)

Since \( g(-x) = -g(x) \), this function is odd.

Final Thoughts

By following these steps and using the definitions of even and odd functions, you can easily classify any function you encounter. Remember that some functions may not fit neatly into either category, so it's essential to check both conditions. If you have specific functions in mind, feel free to share them, and we can analyze them together!