Even and Odd Functions

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

sin x²

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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.

Consider the function given: f(x) = sin(x²). We need to check the behavior of f(-x) compared to f(x).

Calculate f(-x): Substitute -x into the function to get f(-x) = sin((-x)²). Since (-x)² = x², this simplifies to sin(x²).

Compare f(-x) and f(x): We find that f(-x) = sin(x²) = f(x), which matches the condition for an even function.

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).

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주요 개념

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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.

Even and Odd FunctionsIn Exercises 47–62, say whether the function is even, odd, or neither. Give reasons for your answer.sin x²