An even function is a function where the output value is the same regardless of whether the input value is positive or negative. In other words, the function satisfies the equation f(x) = f(-x) for all values of x in the function's domain.
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Even functions are often used to model physical phenomena that exhibit symmetry, such as the motion of a pendulum or the vibration of a guitar string.
The trigonometric functions cosine (cos(x)) and secant (sec(x)) are examples of even functions, while the trigonometric functions sine (sin(x)) and cosecant (csc(x)) are examples of odd functions.
The graph of an even function is symmetric about the y-axis, meaning that the left and right halves of the graph are reflections of each other.
Differentiating an even function results in an odd function, and integrating an even function results in an even function.
Even functions can be represented as a sum of cosine functions, while odd functions can be represented as a sum of sine functions.
Review Questions
Explain the relationship between even functions and the trigonometric functions cosine and secant.
The trigonometric functions cosine (cos(x)) and secant (sec(x)) are examples of even functions. This means that the output values of these functions are the same regardless of whether the input value is positive or negative. For example, cos(x) = cos(-x) and sec(x) = sec(-x) for all values of x in the functions' domains. This symmetry property is a key characteristic of even functions and is exhibited by the cosine and secant functions.
Describe how the graph of an even function is related to its symmetry about the y-axis.
The graph of an even function is symmetric about the y-axis, meaning that the left and right halves of the graph are reflections of each other. This symmetry is a direct consequence of the defining property of even functions, where f(x) = f(-x) for all values of x in the function's domain. As a result, the graph of an even function will appear the same when reflected across the y-axis, indicating that the function's behavior is the same for positive and negative input values.
Analyze the relationship between differentiating and integrating even functions, and how this relates to the representation of even and odd functions using trigonometric functions.
The relationship between differentiating and integrating even functions is an important property. Differentiating an even function results in an odd function, while integrating an even function results in another even function. This is because the derivative of an even function is the product of an odd function (the derivative) and an even function (the original function), which yields an odd function. Conversely, the integral of an odd function is an even function. This property is reflected in the representation of even functions as a sum of cosine functions and odd functions as a sum of sine functions, as cosine is an even function and sine is an odd function.
An odd function is a function where the output value changes sign when the input value changes sign. The function satisfies the equation f(x) = -f(-x) for all values of x in the function's domain.
A periodic function is a function that repeats its values at regular intervals. An even function is a special case of a periodic function where the period is 2.
Symmetry refers to the property of a function where certain transformations, such as reflections or rotations, leave the function unchanged. Even functions exhibit symmetry about the y-axis.