Even Functions
Even functions are a type of mathematical function that satisfy the property f(x) = f(-x) for all x in the function's domain. Visually, even functions are symmetric with respect to the y-axis. Common examples include functions like f(x) = x^2 and f(x) = cos(x). These functions exhibit symmetry and have specific properties that make them useful in various mathematical applications.
3 Key excerpts on "Even Functions"
- F. Xavier Malcata(Author)
- 2020(Publication Date)
- Wiley(Publisher)
...system serves as axis of symmetry for the plot of the former, whereas the origin of coordinates serves as center of symmetry for the plot of the latter. Any function may be written as the sum of an even with an odd function; in fact, (2.1) upon splitting f { x } in half, adding and subtracting f { −x }/2, and algebraically rearranging afterward. Note that f { x } + f { −x } remains unaltered when the sign of x is changed, while f { x } − f {− x } reverses sign when x is replaced by − x ; therefore, (f { x } + f {− x })/2 is an even function, while (f { x } − f {− x })/2 is an odd function. When the value of a function repeats itself at regular intervals that are multiples of some Τ, i.e. f { x + nΤ } = f { x } with n integer, then such a function is termed periodic of period Τ ; a common example is sine and cosine with period 2 π rad, as well as tangent with period π rad (as will be seen below). A (monotonically) increasing function satisfies, whereas a function is called (monotonically). decreasing otherwise, i.e. when ; however, a function may change monotony along its defining range. If y ≡ f { x }, then an inverse function f − 1 { y } may in principle be defined such that f − 1 { f { x }} = x – i.e. composition of a function with its inverse retrieves the original argument of the former. The plot of f − 1 { y } develops around the x ‐axis in exactly the same way the plot of f { x } develops around the y ‐axis; in other words, the curve representing f { x } is to be rotated by π rad around the bisector straight line so as to produce the curve describing f − 1 { y }. Of the several functions worthy of mention for their practical relevance, one may start with absolute value, | x | – defined as (2.2) which turns a nonnegative value irrespective of the sign of its argument; its graph is provided in Fig. 2.1...
- Lawrence S. Leff, Christina Pawlowski-Polanish(Authors)
- 2021(Publication Date)
- Barrons Educational Services(Publisher)
...The graph of an odd function is symmetric to the origin. EXAMPLES The graph in Figure 6.11 describes an odd function since it has origin symmetry. The function f (x) = x 3 − x is an odd function since replacing x with − x gives f (− x) = (− x) 3 − (− x) = − x 3 + x = − f (x). • Typically, a function is neither even nor odd. If g (x) = x 2 + x, then g (− x) = (− x) 2 + (− x) = x 2 − x, so g (− x) ≠ g (x) and function g is not even. Also, because − g (x) = −(x 2 + x) = − x 2 − x ≠ g (− x), function g is not odd. CUBE FUNCTION: f (x) = x 3 If y = x 3, then x and y must have the same sign so that the graph is confined to the. first and third quadrants. Furthermore, there is no restriction on the domain or range. The graph of f (x) = x 3 in Figure 6.12 confirms that the cube function: • has the set of real numbers as its domain and range; • has origin symmetry, so it is an odd function; • includes point (0,0); • increases throughout its domain. FIGURE 6.12 Graph of y = x 3 SQUARE-ROOT FUNCTION: If, then x must be nonnegative, and y is also nonnegative. Therefore, the graph of the square-root function is confined to Quadrant I. The graph of in Figure 6.13 confirms that: • the domain and the range of the square-root function are limited to the sets of all nonnegative real numbers; • the square-root function is neither odd nor even; • the graph of the square-root function includes point (0,0); • the square-root function increases throughout its domain. FIGURE 6.13 Graph of FIGURE 6.14 Graph of RECIPROCAL FUNCTION: If, then x ≠ 0 and x and y must have the same sign so that the graph is confined to the first and third quadrants with no intercepts. The graph of in Figure 6.14 confirms that: • the domain of the reciprocal function is the set of all real numbers except 0, and the range is the set of all nonzero real numbers; • the reciprocal function is odd; • the x- and y- axes are asymptotes of the graph...
eBook - ePub- Nassir H. Sabah(Author)
- 2017(Publication Date)
- CRC Press(Publisher)
...Because a period of an even periodic function is centered about the vertical axis, C n can be expressed as: If we substitute t = − t ′ in the first integral in brackets, this integral becomes ∫ T /2 0 f (− t ′) e j n ω 0 t ′ (− d t ′) = ∫ 0 T / 2 f (− t ′) e j n ω 0 t ′. Changing the dummy integration variable back to t and invoking the property of an even function that f (t) = f (− t), the integral becomes ∫ 0 T / 2 f (t) e j n ω 0 t d t. Substituting in Equation 9.3.1, combining with the second integral, and making use of the relation e jnω0t + e − jnω0t = 2 cos nω 0, we. obtain: C n = 2 T ∫ 0 T l 2 f (t) cos n ω 0 t d t = 2 T Re [ ∫ 0 T / 2 f (t) e − j n ω 0 t d t ] (9.3.2) It follows that for an even. function: a 0 = 2 T ∫ 0 T l 2 f (t) d t, a n = 4 T ∫ 0 T l 2 f (t) cos n ω 0 t d t, and b n = 0 for all n (9.3.3) Odd-Function Symmetry Concept: The FSE of an odd periodic function does not contain an average term nor any cosine terms; its Fourier coefficients can be evaluated over half a period. The reason that the FSE of an odd periodic function does not contain an average term nor any cosine terms is simply that these terms, being even, introduce even components and destroy the odd symmetry of the function. An example of an odd function is the square wave of Figure 9.2.8. The FSE (Equation 9.2.25) consists of sine terms only. However, it is possible that a function that appears to be neither odd nor even becomes odd when the dc component is removed. Examples are the sawtooth waveforms of Figure 9.2.1 and Figure 9.2.4. If the dc component A /2 is subtracted, the function becomes odd. Hence, a function can have an odd ac component but is not odd because of a dc component. If the FSE of an odd periodic function does not contain any cosine terms, a n = 0 and C n is imaginary...
