The article explains even and odd functions and their graphical representations. It also discusses how new even functions can be created from other even or odd functions through addition, multiplication, or composition. Finally, it states that every function can be expressed as the sum of an even and an odd function.
An even function is defined as any function in which the statement f(x) = f(-x) is true for all real values of x. Equivalently, an even function is any function defined for all real values of xe that has reflexive symmetry about the y-axis. Disparity or uniformity of functions is mostly useful in graphing functions.
A function is a relationship that relates the elements of one set of numbers – the domain, to the elements of another set – the range. The relationship is usually defined in terms of a mathematical equation, where if a number from the domain is put into the equation, a single value within the range is given as the answer. For example, for the function f(x) = 3×2 + 1, when x = 2 is the value selected by the domain, f(x) = f(2) = 13. If the domain and range are both from ‘set of real numbers, then the function can be represented graphically by plotting each point (x, f(x)), where the x coordinate is from the domain of the function and the y coordinate is the corresponding value from the interval of the function.
Related to the concept of the even function is the odd function. An odd function is one where the statement f(x) = -f (-x) for all real values of x. When represented graphically, odd functions have rotational symmetry about the origin.
While most functions are neither odd nor even, there are still an infinite number of even functions. The constant function, f(x) = c, where the function has only one value regardless of the value selected from the domain, is an even function. The power functions, f(x) = xn, are even as long as n is an even integer. Of the trigonometric functions, cosine and secant are both even functions, as are the corresponding hyperbolic functions f(x) = cosh(x) = (ex + ex)/2 and f(x) = sech(x) = 2/ ( ex + ex).
New even features can be created from other features known to be even features. Adding or multiplying two even functions will create a new even function. If an even function is multiplied by a constant, the resulting function will be even. Even functions can also be created from odd functions. If two functions known to be odd, such as f(x) = x and g(x) = sin(x), are multiplied together, the resulting function, such as h(x) = x sin(x) will be even.
New even features can also be created from composition. A composition function, such as h(x) = g(f(x)), is one in which the output of one function — in this case f(x) — is used as input to the second function — g(x ). If the innermost function is even, the resulting function will also be even regardless of whether the outer function is even, odd, or neither. The exponential function g(x) = ex, for example, is neither odd nor even, but since cosine is an even function, so is the new function h(x) = echos(x).
A mathematical result states that every function defined for all real numbers can be expressed as the sum of an even and an odd function. If f(x) is any function defined for all real numbers, two new functions can be constructed, g(x) = (f(x) + f(-x))/2 and h(x) = (f ( x) – f(-x))/2. It follows that g(-x) = (f(-x) + f(x))/2 = (f(x) + f(-x))/2 = g(x) and therefore g(x) is an even function. Similarly, h(-x) = (f(-x)-f(x))/2 = – (f(x)-f(-x))/2 = -h(x) so h(x ) is by definition an odd function. If the functions are added, g(x) + h (x) = (f(x)+f(-x))/2 + (f(x)-f(-x))/2 = 2 f( x ) / 2 = f(x). So every function f(x) is the sum of an even and an odd function.
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