Is x¹² an even or odd function? Is sin x² an even or odd function?

Verified step by step guidance

1. Step 1: Recall the definitions of even and odd functions. An even function satisfies f(-x) = f(x), meaning it is symmetric about the y-axis. An odd function satisfies f(-x) = -f(x), meaning it is symmetric about the origin.
2. Step 2: Analyze the function x¹². Substitute -x into the function: f(-x) = (-x)¹². Since raising -x to an even power results in x¹² (because the negative sign disappears), f(-x) = f(x). Therefore, x¹² is an even function.
3. Step 3: Analyze the function sin(x²). Substitute -x into the function: f(-x) = sin((-x)²). Since squaring -x results in x² (because the negative sign disappears), f(-x) = sin(x²). Therefore, sin(x²) satisfies f(-x) = f(x), making it an even function.
4. Step 4: Summarize the results. Both x¹² and sin(x²) are even functions because they satisfy the condition f(-x) = f(x).
5. Step 5: Reflect on the symmetry of these functions. x¹² is symmetric about the y-axis due to the even power, and sin(x²) is symmetric about the y-axis because the argument x² is unaffected by the sign of x.

Key Concepts

Even and Odd Functions

A function is classified as even if it satisfies the condition f(-x) = f(x) for all x in its domain, meaning its graph is symmetric about the y-axis. Conversely, a function is odd if it meets the condition f(-x) = -f(x), indicating that its graph is symmetric about the origin. Understanding these definitions is crucial for determining the nature of specific functions.

Polynomial Functions

Polynomial functions, such as x¹², are expressions that involve variables raised to whole number powers. The degree of the polynomial and the coefficients determine its behavior. In the case of even-degree polynomials, they are always even functions, as substituting -x yields the same result as substituting x.

Trigonometric Functions

Trigonometric functions, like sin(x), have specific properties regarding symmetry. The sine function is an odd function, meaning sin(-x) = -sin(x). When considering sin(x²), we must analyze the argument x², which is always non-negative, affecting the overall symmetry and classification of the function.