There are different types of functions in mathematics that we study. We can determine whether a function is even or odd algebraically or graphically. Even and Odd functions can be checked by plugging in the negative inputs (-x) in place of x into the function f(x) and considering the corresponding output value. Even and odd functions are classified on the basis of their symmetry relations. Even and odd functions are named based on the fact that the power function f(x) = xn is an even function, if n is even, and f(x) is an odd function if n is odd. Show
Let us explore other even and odd functions and understand their properties, graphs, and the use of even and odd functions in integration. A function can be even or odd or both even and odd, or neither even nor odd. Let's explore various examples to understand the concept. What are Even and Odd Functions?Generally, we consider a real-valued function to be even or odd. To identify if a function is even or odd, we plug in -x in place of x into the function f(x), that is, we check the output value of f(-x) to determine the type of the function. Even and odd functions are symmetrical. Let us first understand their definitions. Even and Odd Functions Definition
Even and Odd Functions in TrigonometryIn this section, we will segregate trigonometric functions as even and odd functions. We have six trigonometric ratios (sine, cosine, tangent, cotangent, cosecant, and secant). These trigonometric ratios give positive values in different quadrants for various measures of angles. In the first quadrant (where x and y coordinates are all positive), all six trigonometric ratios have positive values. In the second quadrant, only sine and cosecant are positive. In the third quadrant, only tangent and cotangent are positive. In the fourth quadrant, only cosine and secant are positive. Based on these signs, we will categorize them as even and odd functions. If a trigonometric ratio is even or odd can be checked through a unit circle. An angle measured in anticlockwise direction is a positive angle whereas the angle measured in the clockwise direction is a negative angle.
Integral Properties of Even and Odd FunctionsThe integral of a function gives the area below the curve. We use properties of even and odd functions while solving definite integrals. For that, we need to know the limits of the integral and the nature of the function. If the function is even or odd, and the interval is [-a, a], we can apply the following two rules:
Even and Odd Functions GraphLet us now see how even and odd functions behave graphically. The graph of an even function is symmetric with respect to the y-axis. In other words, the graph of an even function remains the same after reflection about the y-axis. For any two opposite input values of x, the function value will remain the same all along the curve. Whereas the graph of an odd function is symmetric with respect to the origin. In other words, the graph of an odd function is at the same distance from the origin but in opposite directions. For any two opposite input values of x, the function has opposite y values. Here are a few examples of even and odd functions. Properties of Even and Odd Functions
Important Notes on Even and Odd Functions
Related Topics on Even and Odd Functions
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FAQs on Even and Odd FunctionsA function f(x) is even if f(-x) = f(x), for all values of x in D(f) and it is odd if f(-x) = -f(x), for all values of x. The graph even function is symmteric with respect to the y-axis and the graph of an odd function is symmetric about the origin. A real-valued function f(x) is said to be both even and odd if it satisifies f(-x) = f(x) and f(-x) = -f(x) for all values of x in the domain of the function f(x). How to Determine Even and Odd Functions Algebraically?To check if the function is even or odd algebraically, we check whether f(-x) = f(x) or f(-x) = -f(x) for all values of x, respectively. If we substitute x with -x in the function and the value of the function becomes negative, then the function is called an odd function. If we substitute x with -x in the function and the value of function does not change then the function is known even function. If a function f is an even function, then what type of symmetry does the graph of f have?Even and Odd functions show different types of symmetries. Even functions have line symmetry. The line of symmetry is the y-axis. Even functions are the function in which when we substitute x by -x, then the value of the function for that particular x does not change. The graph of the even function behaves equally for all the points on x-axis that are on the left of the origin as well as the right of the origin. Which Trigonometric Functions are Even and Odd Functions?In trigonometry, cosθ and secθ are even functions, and sinθ, cosecθ, tanθ, cotθ are odd functions. Different trigonometric ratios have positive-negative values in different quadrants. Hence, using a unit circle we can see that cosθ and secθ are even functions and the remaining four trigonometric ratios are odd. How to Identify Even and Odd Functions Graphically?The graph of an even function is symmetric with respect to the y-axis. The graph of an odd function is symmetric with respect to the origin. The graph of an even function remains the same after reflection about the y-axis. The graph of an odd function is at the same distance from the origin but in opposite directions. Using the same criteria, we can identify even and odd functions graphically. If f and g are both even functions, is f+g an even and odd function?If f and g are even functions, then f + g is an even function. Consider a function h(x) = f(x) + g(x). Replace x with -x in h(x), h(-x) = f(-x) + g(-x) = f(x) + g(x) = h(x). Since f(x) and g(x) are even, f(-x) = f(x) and g(-x) = g(x). Hence, f + g is an even function. It is not both even and odd function. |