In Exercises 9–16, determine whether the function is even, odd, or neither.


𝔂 = x² + 1

Verified step by step guidance

1. To determine if a function is even, odd, or neither, we need to analyze its symmetry properties. A function y = f(x) is even if f(-x) = f(x) for all x in the domain, and it is odd if f(-x) = -f(x) for all x in the domain.
2. Start by substituting -x into the function y = x² + 1 to find f(-x). This gives us f(-x) = (-x)² + 1.
3. Simplify the expression f(-x) = (-x)² + 1. Since (-x)² = x², we have f(-x) = x² + 1.
4. Compare f(-x) with f(x). We have f(-x) = x² + 1 and f(x) = x² + 1. Since f(-x) = f(x), the function is even.
5. Conclude that the function y = x² + 1 is even because it satisfies the condition f(-x) = f(x) for all x in its domain.

Key Concepts

Even Functions

A function is considered 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 with respect to the y-axis. For example, the function f(x) = x² is even because substituting -x yields the same result as substituting x.

Odd Functions

A function is classified as 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 with respect to the origin. An example of an odd function is f(x) = x³, where substituting -x results in the negative of the function's value.

Neither Even Nor Odd Functions

A function is neither even nor odd if it does not satisfy the conditions for either classification. This means that the function's graph lacks symmetry with respect to both the y-axis and the origin. For instance, the function f(x) = x² + 1 is neither even nor odd, as it does not exhibit the required symmetries.