How can I tell if a graph is even or odd?

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Determining if a Graph is Even or Odd

To determine whether a graph is even, odd, or neither, you need to analyze its symmetry properties concerning the y-axis and the origin. Here’s a detailed breakdown:

1. Definitions

  • Even Function: A function is considered even if its graph is symmetric about the y-axis. This means that for every point (x, f(x)) on the graph, there exists a corresponding point (-x, f(x)) that is also on the graph.
  • Odd Function: A function is considered odd if its graph has rotational symmetry about the origin. This means that for every point (x, f(x)) on the graph, the point (-x, -f(x)) must also be present.

2. Steps to Determine Evenness or Oddness

  1. Check for Evenness:
    • Take your function, f(x).
    • Evaluate f(-x) and see if it equals f(x).
    • If f(-x) = f(x), then the function is even.
  2. Check for Oddness:
    • Using the same function, f(x), evaluate f(-x).
    • Check if f(-x) = -f(x).
    • If this condition holds true, then the function is odd.

3. Examples

  • Example of an Even Function: Consider f(x) = x².
    • f(-x) = (-x)² = x² = f(x) → This confirms f(x) is even.
  • Example of an Odd Function: Now take f(x) = x³.
    • f(-x) = (-x)³ = -x³ = -f(x) → This confirms f(x) is odd.

4. Conclusion

By following these steps, you can efficiently determine if a graph is even, odd, or neither. This understanding helps greatly in graph analysis and provides insights into the properties and behaviors of different functions.

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