Exploring the Domain and Range of the Cosine Function (fx cos x)

The cosine function, denoted as cosx, is a fundamental trigonometric function that plays a crucial role in mathematics and its applications. This article will delve into the concept of the domain and range of fx cosx, providing a comprehensive understanding of its characteristics and behaviors.

Understanding the Domain of fx cosx

The domain of a function is the set of all possible input values (x-values) for which the function is defined and produces real output values. For the cosine function, the domain is all real numbers, which means x can take on any value from negative infinity to positive infinity. This is generally expressed as:

Domain of cosx { -∞, ∞ }

Exploring the Range of fx cosx

The range of a function is the set of all possible output values (f(x)-values) that the function can produce. For the cosine function, the output values lie between -1 and 1. This makes the range of cosx as:

Range of cosx [-1, 1]

Graphical Representation

The graph of the cosine function is a periodic wave that oscillates between -1 and 1. Here's a simplified representation of how the domain and range are represented graphically:

The graph of cosx starts at 1 at 0 radians, decreases to -1 at π radians, and then increases back to 1 at 2π radians. This pattern repeats indefinitely in both directions.

The Domain and Range of Cosine in the Complex Plane

The cosine function can be extended to the complex plane, where x can be a complex number. In this context, the cosine function is defined using Euler's formula:

cos z (e^{iz} e^{-iz})/2

Here, z is a complex number. This definition ensures that the cosine function is always defined for any complex number z. This means that the domain of the extended cosine function includes all complex numbers:

Domain of cosz All complex numbers, mathbb{C}

Simplified Explanation

For practical purposes, when working with real numbers, the domain of cosx is all real numbers, and the range is limited to between -1 and 1. This can be expressed as:

Domain of cosx [-∞, ∞]