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Find the range of $\{\tan x + \sin x\}$ where $\{.\}$ represents the fractuonal part of function

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Domain and range of square root functions: equations

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Q. Find the range of

\{\tan x + \sin x\}

where

\{.\}

represents the fractuonal part of function

Definition of Fractional Part Function: We need to understand the definition of the fractional part function. The fractional part of a number $x$ , denoted as $\{x\}$ , is the part of $x$ that remains after subtracting the largest integer less than or equal to $x$ . For example, if $x = 3.7$ , then $\{x\} = 0.7$ . The range of the fractional part function is always $[0, 1)$ , because it represents the non-integer part of a number.
Consideration of $tan(x) + sin(x)$ : Now let's consider the function $tan(x) + sin(x)$ . The range of $sin(x)$ is $[-1, 1]$ , and the range of $tan(x)$ is all real numbers, except where $cos(x) = 0$ (which would make $tan(x)$ undefined). However, since we are looking for the fractional part, we are only interested in the values of $tan(x) + sin(x)$ modulo $1$ .
Range of $tan(x) + sin(x)$ : The sum of $tan(x)$ and $sin(x)$ can be any real number, because $tan(x)$ can take any real value. However, no matter what the sum is, the fractional part will always be between $0$ and $1$ , exclusive of $1$ . This is because the fractional part function only takes the non-integer part of a number.
Conclusion of Range: Therefore, the range of the fractional part of the function $\tan(x) + \sin(x)$ is $[0, 1)$ . This is because the fractional part strips away the integer part and leaves a value between $0$ (inclusive) and $1$ (exclusive), regardless of the integer part of the original sum.

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