If f(x)=(x(sinx+tanx))/([(x+π)/(π)]-(1)/(2)), where [ ] denotes greatest integer function, then

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If f(x)=(x(sinx+tanx))/([(x+π)/(π)]-(1)/(2)), where [ ]...

If $f(x)=\frac{x(\sin x+\tan x)}{\left[\frac{x+\pi}{\pi}\right]-\frac{1}{2}}$, where $[$ ] denotes greatest integer function, then (A) $f(x)$ is an odd function if $x=n \pi$ (B) $f(x)$ is an even function if $x \neq n \pi$ (C) $f(x)$ is an odd function if $x \neq n \pi$ (D) $f(x)$ is an even function if $x=n \pi$

If $f(x)=\frac{x(\sin x+\tan x)}{\left[\frac{x+\pi}{\pi}\right]-\frac{1}{2}}$, where $[$ ] denotes greatest
integer function, then
(A) $f(x)$ is an odd function if $x=n \pi$
(B) $f(x)$ is an even function if $x \neq n \pi$
(C) $f(x)$ is an odd function if $x \neq n \pi$
(D) $f(x)$ is an even function if $x=n \pi$

Key Concepts

Even and Odd Functions

Even and odd functions are categories based on symmetry: an even function satisfies f(–x) = f(x) and is symmetric about the y?axis, while an odd function satisfies f(–x) = –f(x) and is symmetric about the origin. Recognizing whether a function is even or odd simplifies analysis of its behavior and can reveal properties like cancellation in integrals over symmetric intervals.

Greatest Integer Function

The greatest integer function, also known as the floor function, assigns to each real number the largest integer that is less than or equal to it. This function is discontinuous at integer points, and its application within a composite function can introduce piecewise behavior and affect symmetries, making careful consideration of its domain and jumps essential.

Trigonometric Function Properties

Trigonometric functions possess inherent symmetric properties; for example, the sine and tangent functions are odd functions. This means that individually they satisfy f(–x) = –f(x). When combined algebraically with other functions, these parity properties can contribute to the overall symmetry of the composite function.

Function Composition and Domain Analysis

When composing functions, especially when one component is piecewise or discontinuous like the floor function, it is important to analyze how the overall function behaves over different regions of its domain. This includes checking for points of discontinuity or undefined expressions, as well as verifying the transformation properties that affect symmetry, such as the effects of translations or scalings.

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