Even functions are functions that satisfy the property f(x) = f(-x) for all x in their domain. This means that the graph of an even function is symmetric with respect to the y-axis. Here are some examples:
Examples of Even Functions
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Polynomial Functions with Even Powers: Any polynomial function where all the exponents are even numbers is an even function. Examples include:
- f(x) = x2
- f(x) = x4
- f(x) = x6
- Generally, f(x) = xn where n is an even integer.
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The Absolute Value Function: The absolute value function, denoted as f(x) = |x|, is an even function because |x| = |-x| for all x.
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Cosine Function: The cosine function, f(x) = cos(x), is a classic example of an even function. This is because cos(x) = cos(-x) for all x.
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Cosine of a Multiple of x: Functions of the form f(x) = cos(nx), where n is a constant, are even. For example:
- cos(2x)
- cos(3x)
- cos(nx)
Explanation
To determine if a function is even, you can substitute -x for x in the function's expression. If the result is the same as the original function, then the function is even.
Example:
Let's check if f(x) = x2 is an even function:
f(-x) = (-x)2 = x2 = f(x)
Since f(-x) = f(x), the function f(x) = x2 is even.
