Introduction to Finding the Domain and Range of the Function

This article will delve into the process of determining the domain and range of the real-valued function [ f(x) frac{1 - tan x}{1 - tan x} ]. We will discuss the properties of the tangent function and the behavior of rational functions to help us identify the values of (x) and (y) for which the function is defined and takes real values, respectively.

Understanding the Function

Let's examine the function [ f(x) frac{1 - tan x}{1 - tan x} ]. This function is a rational function where the numerator and the denominator are both linear in (tan x). Understanding the domain and range of functions like this involves delving into the behavior of the tangent function, which is central to many aspects of trigonometry.

Determining the Domain

Exclusion of Values Where (tan x) is Undefined

The tangent function (tan x) is undefined at (x frac{pi}{2} npi) for any integer (n). This is a crucial point to exclude from the domain of (f(x)), as division by zero is not allowed.

Additionally, the function (f(x)) is undefined when the denominator is zero, which occurs when (tan x 1). This happens at (x frac{pi}{4} npi) for any integer (n).

Therefore, the domain of (f(x)) can be mathematically represented as:

[ D left{ x in mathbb{R} mid x eq frac{pi}{2} npi text{ and } x eq frac{pi}{4} npi, n in mathbb{Z} right} ]

Visualizing the Domain

A quick review of the domain reveals that the function (f(x)) is not defined for multiples of (frac{pi}{2}) and (frac{pi}{4}), indicating periodic discontinuities.

Exploring the Range

Manipulating the Function

Let (y f(x)). Then the equation becomes:[ y frac{1 - tan x}{1 - tan x} ]

Rearranging, we find:[ y(1 - tan x) 1 - tan x ]

This simplifies to:[ y(1 - tan x) - (1 - tan x) 0 ]

Further simplification yields:[ (y - 1)(1 - tan x) 0 ]

Hence, either (y - 1 0) or (1 - tan x 0), implying (y 1) or (tan x 1).

Ensuring Validity of the Output

The expression (frac{y - 1}{y 1}) is defined for all (y eq -1). Therefore, the function (f(x)) does not achieve the value (y -1).

Thus, the range of (f(x)) is:

[ R left{ y in mathbb{R} mid y eq -1 right} ]

Conclusion

The domain and range of the function (f(x) frac{1 - tan x}{1 - tan x}) are:

Domain: (D left{ x in mathbb{R} mid x eq frac{pi}{2} npi text{ and } x eq frac{pi}{4} npi, n in mathbb{Z} right})

Range: (R left{ y in mathbb{R} mid y eq -1 right})

Acknowledgments

For further insights into similar problems and to deepen your understanding of trigonometric functions and rational functions, the following resources may prove beneficial:

Understanding Trigonometric Identities and Functions - [Link to a detailed article]

Exploring Rational Functions - [Link to a related article]

Leveraging the above-provided equations and concepts can help in enhancing your approach to identifying the domain and range of complex functions. This knowledge is foundational for more advanced topics in mathematics and engineering.