Which of the following functions is even?
In mathematics, 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 an even function is symmetric with respect to the y-axis. In this article, we will explore several functions and determine which one is even among them.
The first function we will consider is f(x) = x^2. To determine if this function is even, we need to check if f(-x) = f(x) holds true. By substituting -x into the function, we get f(-x) = (-x)^2 = x^2. Since f(-x) = f(x), we can conclude that f(x) = x^2 is an even function.
The second function is g(x) = x^3. To determine if this function is even, we will again check if g(-x) = g(x) holds true. By substituting -x into the function, we get g(-x) = (-x)^3 = -x^3. Since g(-x) ≠ g(x), we can conclude that g(x) = x^3 is not an even function.
The third function is h(x) = |x|. To determine if this function is even, we will check if h(-x) = h(x) holds true. By substituting -x into the function, we get h(-x) = |-x| = |x|. Since h(-x) = h(x), we can conclude that h(x) = |x| is an even function.
The fourth function is j(x) = x^4. To determine if this function is even, we will check if j(-x) = j(x) holds true. By substituting -x into the function, we get j(-x) = (-x)^4 = x^4. Since j(-x) = j(x), we can conclude that j(x) = x^4 is an even function.
In conclusion, among the functions provided, f(x) = x^2, h(x) = |x|, and j(x) = x^4 are even functions. They all satisfy the condition f(-x) = f(x) for all x in their respective domains.
