Even and Odd Functions


In Exercises 47–62, say whether the function is even, odd, or neither. Give reasons for your answer.


sin x²

Verified step by step guidance

1. Understand the definitions: A function f(x) is even if f(-x) = f(x) for all x in the domain, and odd if f(-x) = -f(x) for all x in the domain.
2. Consider the function given: f(x) = sin(x²). We need to check the behavior of f(-x) compared to f(x).
3. Calculate f(-x): Substitute -x into the function to get f(-x) = sin((-x)²). Since (-x)² = x², this simplifies to sin(x²).
4. Compare f(-x) and f(x): We find that f(-x) = sin(x²) = f(x), which matches the condition for an even function.
5. Conclude: Since f(-x) = f(x), the function sin(x²) is even. It does not satisfy the condition for an odd function, which would require f(-x) = -f(x).

Key Concepts

Even Functions

A function is classified as 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. Common examples include polynomial functions with only even powers, such as f(x) = x².

Odd Functions

A function is considered 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. Typical examples include polynomial functions with only odd powers, such as f(x) = x³.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, have specific properties regarding their symmetry. The sine function is an odd function, meaning sin(-x) = -sin(x). Understanding these properties is crucial when analyzing functions like sin(x²) to determine their evenness or oddness.