Even function
An even function is a type of function in mathematics that exhibits a particular symmetry property
An even function is a type of function in mathematics that exhibits a particular symmetry property. If a function f(x) is even, it means that it has symmetry about the y-axis. This can be visually observed by noticing that if you reflect the graph of the function across the y-axis, it remains unchanged.
Mathematically, a function f(x) is even if and only if f(x) = f(-x) for all x in the domain of the function. This means that if you replace x with its opposite (-x) in the function, the result should be the same as the original function.
In terms of even functions, some key characteristics to note are:
1. The graph of an even function is symmetric about the y-axis. This means that for any given point (x, y) on the graph, the point (-x, y) should also lie on the graph.
2. The domain of an even function can be both positive and negative real numbers or any other interval symmetric about the origin.
3. If a function is even, its graph can be described using only the right half or left half of the graph, since they are mirror images of each other.
Examples of even functions include:
1. f(x) = x^2, where f(-x) = (-x)^2 = x^2. The graph of this function is a symmetric parabola opening upwards.
2. f(x) = cos(x), where f(-x) = cos(-x) = cos(x). The graph of this function is a symmetric curve oscillating around the x-axis.
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