even function
f(-x)=f(x) for all values of x in the domain of the function. The graph of the function will be symmetric across the y-axis.
In mathematics, a function f(x) is said to be even if f(-x) = f(x) for all values of x in the domain of f. This means that the function has symmetry about the y-axis. Alternatively, we can say that a function is even if its graph is symmetric with respect to the y-axis.
Geometrically, an even function is a mirror image of itself. For example, the function f(x) = x² is even because f(-x) = (-x)² = x² = f(x) for all values of x. The graph of this function is shown below, and we can observe the symmetry about the y-axis.
Even functions have some interesting properties. For instance, if f(x) is an even function, then the integral of f(x) from -a to a (where a is a positive number) is twice the integral of f(x) from 0 to a. This is because the function is symmetric about the y-axis, and so the areas above and below the x-axis cancel out.
In summary, an even function is a function whose graph is symmetric about the y-axis. It is defined as f(x) = f(-x) for all values of x in the domain of f. Even functions have some important properties, including symmetry and special integrals.
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