Question

If \([.]\) represents greatest integer function, then which of the following, is identical to \(\mathrm{f}\) of \((\mathrm{x})\) for \(\mathrm{x} \in(\pi, 2 \pi)\) (A) \(\operatorname{sgn}(\mathrm{x})\) (B) \(\frac{x}{x}\) (C) \(\sin \pi[\mathrm{x}]\) (D) \(\frac{\sin x}{\sin x}\)

Short answer

A. \(\operatorname{sgn}(x)\) B. \(\frac{x}{x}\) C. \(\sin(\pi[x])\) D. \(\frac{\sin x}{\sin x}\) Answer: (B) and (D) Explanation: In the interval \((\pi, 2\pi)\), both expressions in options B and D simplify to \(1\), which matches with \(f(x)\) calculated in the same interval.

Step-by-step solution

Option A - Check \(\operatorname{sgn}(x)\)

The sign function (sgn) is defined as: $$\operatorname{sgn}(x) = \begin{cases} \-1, & \text{if }x < 0, \\ \ 0, & \text{if }x = 0, \\ \ 1, & \text{if }x > 0. \end{cases}$$ Since \(x\) is always positive in the given interval \((\pi, 2\pi)\), \(\operatorname{sgn}(x) = 1\). We cannot conclude anything about matching with \(f(x)\) yet, so let's analyze the other options.

Option B - Check \(\frac{x}{x}\)

The expression \(\frac{x}{x}\) is equal to \(1\) for all \(x \neq 0\), and since \(x\) is never equal to \(0\) in the given interval \((\pi, 2\pi)\), this expression is also equal to \(1\). Again, we cannot conclude anything about matching with \(f(x)\), so let's analyze the other options.

Option C - Check \(\sin(\pi[x])\)

Let's consider what happens to \(\sin(\pi[x])\) in the interval \((\pi, 2\pi)\). As \(x\) ranges from \(\pi\) to \(2\pi\), all values of \(\pi[x]\) will be the integer multiples of \(\pi\). In other words, \(\pi[x] = k\pi\) for some integer \(k\). The sine of an integer multiple of \(\pi\) is always \(0\). So, \(\sin(\pi[x]) = 0\) in the given interval. This doesn't seem to match with \(f(x)\), so let's analyze the last option.

Option D - Check \(\frac{\sin x}{\sin x}\)

The expression \(\frac{\sin x}{\sin x}\) simplifies to \(1\), whenever \(\sin x \neq 0\). Since \(x\) lies in the interval \((\pi, 2\pi)\), \(\sin x\) is never equal to \(0\). Thus, this expression is also equal to \(1\) for all \(x\) in the given interval. #Conclusion# After analyzing all four options, we can conclude that options B and D match with \(f(x)\) for the given interval \((\pi, 2\pi)\), as they both give the same value of \(1\). Therefore, the answer is \(\boxed{\text{(B) and (D)}}\).