How to Analyze the Properties of the Function f(x) cos(tan(x))?

To determine the properties of the function f(x) cos(tan(x)), we can examine several critical aspects, including its domain, range, periodicity, continuity, and behavior at specific points. This detailed breakdown will provide insights into the characteristics of this function.

1. Domain

The function tan(x) is defined everywhere except at its vertical asymptotes, which occur at x (2k 1)π/2 for any integer k. Therefore, the domain of f(x) cos(tan(x)) is all real numbers except for these points. Mathematically, this can be expressed as:

x ∈ ?, x ≠ (2k 1)π/2, k ∈ ?

2. Range

The output of tan(x) can take any real value as x approaches the vertical asymptotes. The function cos(y) oscillates between -1 and 1 for any real number y. Therefore, the range of f(x) cos(tan(x)) is between -1 and 1:

cos(tan(x)) ∈ [-1, 1]

3. Periodicity

The function tan(x) is periodic with a period of π but cos(y) is periodic with a period of 2π. This results in the function cos(tan(x)) not having a simple periodicity, due to the nature of tan(x).

However, f(x) exhibits repeating behavior within intervals that do not include the points where tan(x) is undefined. This means that the function does not have a well-defined period but it shows a pattern within intervals of the form (kπ - π/2, kπ π/2) for integer k.

4. Continuity

The function f(x) cos(tan(x)) is continuous wherever tan(x) is defined. Thus, it is continuous on intervals of the form (kπ - π/2, kπ π/2) for integer k.

5. Behavior at Specific Points

At x 0, the function evaluates to:

f(0) cos(tan(0)) cos(0) 1

As x approaches π/2, tan(x) approaches infinity, causing cos(tan(x)) to oscillate between -1 and 1.

6. Derivative and Critical Points

To further analyze the function, you can take its derivative:

f'(x) -sin(tan(x)) * sec2(x)

This derivative helps identify where the function increases or decreases. By setting the derivative to zero, you can find the critical points of the function.

Summary

The function f(x) cos(tan(x)) has a complex structure due to the properties of both tan(x) and cos(y). It is defined on intervals excluding vertical asymptotes of tan(x), oscillates between -1 and 1, and does not have a simple periodicity due to the behavior of tan(x).

For a more detailed analysis, consider plotting specific intervals or using numerical methods to investigate local maxima and minima.