How do you use possible symmetry to determine whether the graph is the graph of an even function, an odd function, of a function that is neither even or odd?
To determine if a graph represents an even function, an odd function, or a function that is neither even nor odd, you can use the concept of symmetry.
1. Even Function:
An even function has symmetry with respect to the y-axis, meaning that if you fold the graph in half along the y-axis, the two halves will be identical. To determine if a graph represents an even function:
- Check if the function is defined on both positive and negative x-values.
- Evaluate f(x) = f(-x) for various x-values. If f(x) = f(-x) holds true for every x in the domain, the graph represents an even function.
2. Odd Function:
An odd function has symmetry with respect to the origin (0,0), meaning that if you rotate the graph 180 degrees around the origin, the resulting image will be identical to the original graph. To determine if a graph represents an odd function:
- Check if the function is defined on both positive and negative x-values.
- Evaluate f(x) = -f(-x) for various x-values. If f(x) = -f(-x) holds true for every x in the domain, the graph represents an odd function.
3. Neither Even nor Odd:
If a graph does not exhibit symmetry with respect to the y-axis or the origin, it represents a function that is neither even nor odd.
Remember, to apply these concepts, you need to consider the entire domain of the function and ensure it is defined for both positive and negative x-values. By evaluating the function's values at positive and negative x-values, you can determine the symmetry and classify the graph accordingly.