Even Functions In Mathematics: Definition, Examples, And Properties

Even Functions

f(-x) = f(x)

An even function is a kind of function that satisfies a certain property. Specifically, a function f(x) is said to be even if it has the property that f(-x) = f(x) for all values of x. This means that the function is symmetric about the y-axis.

For example, consider the function f(x) = x^2. This function has the property that f(-x) = f(x) for all values of x. To see this, we can substitute -x for x in the expression for f(x) and simplify:

f(-x) = (-x)^2 = x^2 = f(x)

Thus, f(x) is an even function.

Similarly, the function g(x) = |x| is also an even function. To see this, we can consider that g(x) is defined as:

g(x) = x if x >= 0
g(x) = -x if x < 0 If we substitute -x for x, we get: g(-x) = -x if -x >= 0 (which is equivalent to x <= 0) g(-x) = x if -x < 0 (which is equivalent to x > 0)

In either case, we see that g(-x) = |x| = g(x). This shows that g(x) is an even function.

Even functions have some important properties that are useful in mathematics. For example, if an even function is defined on the interval [-a, a], then its integral over that interval can be expressed as twice the integral over [0, a]. This is because the function is symmetric about the y-axis and the areas on either side cancel out. Additionally, the Fourier series of an even function contains only cosine terms and no sine terms. Overall, even functions play an important role in various areas of mathematics and are useful tools for solving problems.

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