f(x)=sin[x]+[sinx], 0<x<(π)/(2), where [] represents the greatest integer function can also be r

step 1 In this question, we have to define the function, that is, I can say that fx equals to sine of g

F(x)=sin[x]+[sinx], 0<x<(π)/(2), where [] represents the...

$f(x)=\sin [x]+[\sin x], 0<x<\frac{\pi}{2}$, where [] represents the greatest integer function can also be represented as (a) $\left\{\begin{array}{ll}0 \quad, 0<\mathrm{x}<1 \\ 1+\sin 1,1 \leq \mathrm{x}<\frac{\pi}{2}\end{array}\right.$ (b) $\left\{\begin{array}{l}\frac{1}{\sqrt{2}} \quad, 0<\mathrm{x}<\frac{\pi}{4} \\ 1+\frac{1}{2}+\frac{1}{\sqrt{2}}+\frac{\sqrt{3}}{2}, \frac{\pi}{4} \leq \mathrm{x}<\frac{\pi}{2}\end{array}\right.$ (c) $\left\{\begin{array}{ll}0 & , 0<\mathrm{x}<1 \\ \sin 1, & 1 \leq \mathrm{x}<\frac{\pi}{2}\end{array}\right.$ (d) $\left\{\begin{array}{ll}0 & , \quad 0<\mathrm{x}<\frac{\pi}{4} \\ 1 & , \frac{\pi}{4}<\mathrm{x}<1 \\ \sin 1 & , 1 \leq \mathrm{x}<\frac{\pi}{2}\end{array}\right.$

$f(x)=\sin [x]+[\sin x], 0<x<\frac{\pi}{2}$, where [] represents the greatest integer function can also be represented as
(a) $\left\{\begin{array}{ll}0 \quad, 0<\mathrm{x}<1 \\ 1+\sin 1,1 \leq \mathrm{x}<\frac{\pi}{2}\end{array}\right.$
(b) $\left\{\begin{array}{l}\frac{1}{\sqrt{2}} \quad, 0<\mathrm{x}<\frac{\pi}{4} \\ 1+\frac{1}{2}+\frac{1}{\sqrt{2}}+\frac{\sqrt{3}}{2}, \frac{\pi}{4} \leq \mathrm{x}<\frac{\pi}{2}\end{array}\right.$
(c) $\left\{\begin{array}{ll}0 & , 0<\mathrm{x}<1 \\ \sin 1, & 1 \leq \mathrm{x}<\frac{\pi}{2}\end{array}\right.$
(d) $\left\{\begin{array}{ll}0 & , \quad 0<\mathrm{x}<\frac{\pi}{4} \\ 1 & , \frac{\pi}{4}<\mathrm{x}<1 \\ \sin 1 & , 1 \leq \mathrm{x}<\frac{\pi}{2}\end{array}\right.$

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Key Concepts

Greatest Integer Function

The greatest integer function (or floor function) maps any real number to the largest integer that is less than or equal to it. This function results in a step-like graph that remains constant between integers and jumps at integer values, which is a key behavior when analyzing functions involving it.

Piecewise Functions

Piecewise functions are defined by different rules or formulas over different parts of their domains. Recognizing and constructing piecewise definitions is crucial when working with functions that change behavior based on intervals, such as those incorporating the floor function.

Trigonometric Functions

Trigonometric functions, like the sine function, have well-defined ranges and properties over specific intervals. Their continuous and predictable behavior, especially in intervals where they are monotonic, is essential to analyze when they are used in combination with discontinuous functions like the floor function.

Domain Partitioning

Domain partitioning involves breaking the function’s domain into different regions over which the function can be expressed in a simpler form. This technique is particularly important when combining functions with abrupt changes, as it allows for a clearer analysis and representation of the overall function.

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