Which of the following is an even function? This question often appears in mathematics courses, particularly in the study of functions and their properties. An even function is a mathematical function that satisfies the condition f(-x) = f(x) for all x in its domain. In this article, we will explore the concept of even functions, their characteristics, and some examples to help you identify which of the given options is an even function.
An even function is characterized by its symmetry about the y-axis. When a graph of an even function is reflected across the y-axis, the resulting graph is identical to the original. This symmetry is a key feature that distinguishes even functions from odd functions, which exhibit symmetry about the origin.
To determine whether a given function is even, we can apply the definition mentioned earlier. Let’s consider the following examples:
1. f(x) = x^2
2. g(x) = x^3
3. h(x) = x^4
4. j(x) = x^5
We need to check if any of these functions satisfy the condition f(-x) = f(x) for all x in their respective domains.
For f(x) = x^2, we have f(-x) = (-x)^2 = x^2. Since f(-x) = f(x), this function is even.
For g(x) = x^3, we have f(-x) = (-x)^3 = -x^3. Since f(-x) ≠ f(x), this function is not even.
For h(x) = x^4, we have f(-x) = (-x)^4 = x^4. Since f(-x) = f(x), this function is even.
For j(x) = x^5, we have f(-x) = (-x)^5 = -x^5. Since f(-x) ≠ f(x), this function is not even.
Based on our analysis, the even functions from the given options are f(x) = x^2 and h(x) = x^4. Therefore, the correct answer to the question “Which of the following is an even function?” is either f(x) = x^2 or h(x) = x^4, depending on the specific options provided.
Understanding the concept of even functions is crucial in various fields of mathematics and its applications. By recognizing the symmetry and properties of even functions, we can solve problems more efficiently and gain a deeper understanding of mathematical relationships.
