Determine the nature of the function f(x)=x sin^2 x+tan(x^5)+log((2-x)/(2+x))

Name: Problem 159, Chapter 1: Differential Calculus Booster with Problems and Solutions for JEE Main and Advanced
Uploaded: 2022-02-26T18:40:47-08:00
Duration: 1 min 51 s
Channel: Varsha Aggarwal
Description: Determine the nature of the function f(x)=x sin^2 x+tan(x^5)+log((2-x)/(2+x))

step 1 equal x sine square x plus 10 x to the power 5 plus lobe 2 minus x upon 2 plus x log 2 minus x u

Determine the nature of the function f(x)=x sin^2...

Key Concepts

Composite Function Analysis

The function is built by combining and composing several elementary functions—such as polynomial, trigonometric, and logarithmic functions. Analyzing how these functions interact helps in determining the overall character of the function, including its domain, symmetry, and points of discontinuity. This concept is key to unraveling the behavior of more complex functions constructed from simpler ones.

Continuity and Differentiability

This concept examines where the function behaves 'nicely' and where it might have breaks or singularities. For the given function, the continuity and differentiability are affected by points where the logarithmic and tangent functions are undefined. Recognizing these points and understanding their impact on the behavior of the overall function is essential for a thorough analysis.

Function Domain

This concept involves determining the set of input values for which the function is defined. In the given function, restrictions are imposed by the logarithmic term, which requires its argument to be positive, and by the tangent term, which is undefined when its cosine denominator is zero. The overall domain is the intersection of the individual domains of these components, making it a crucial concept for understanding the function.

Symmetry Properties

Symmetry analysis includes checking whether a function is even, odd, or neither. By examining the individual components in the function—where terms like x sin²x, tan(x?), and the logarithmic expression each exhibit odd symmetry—one can determine that the overall function is odd. This means that f(–x) = –f(x), which often simplifies further analysis and graphing of the function.

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