The real valued function f(x) = cosec⁻¹ x / √(x-[x]), where [x] denotes the greatest integer less th

Q: Wait, why is $\text{cosec}^{-1} x$ defined for $|x| \ge 1$?

Great question! Remember that $\text{cosec} \theta = \frac{1}{\sin \theta}$. Since the range of $\sin \theta$ is $[-1, 1]$, the range of $\text{cosec} \theta$ is $(-\infty, -1] \cup [1, \infty)$. Therefore, for the inverse function $\text{cosec}^{-1} x$ to exist, the input $x$ must lie within this range, which is $|x| \ge 1$.

Q: Why do we exclude integers in the denominator?

The denominator is $\sqrt{x-[x]}$. We know $x-[x]$ is the fractional part of $x$, denoted as $\{x\}$. The fractional part $\{x\}$ is always in the interval $[0, 1)$. If $x$ is an integer, $\{x\} = 0$. Since the square root of zero is zero, and we cannot divide by zero, we must exclude all integers to ensure the denominator is strictly positive.

Q: Why is the final domain 'all non-integers except the interval $[-1, 1]$'?

Let's synthesize. The numerator condition gives $x \in (-\infty, -1] \cup [1, \infty)$. The denominator condition gives $x \notin \mathbb{Z}$. When we intersect these, we take the union $(-\infty, -1] \cup [1, \infty)$ and remove all integers. Since $-1$ and $1$ are integers, they are removed, turning the closed brackets into open ones. Thus, we get $(-\infty, -1) \cup (1, \infty)$ excluding all other integers, which is exactly the set of all non-integers outside the interval $[-1, 1]$.

Question

The real valued function $f(x) = \frac{	ext{cosec}^{-1} x}{\sqrt{x-[x]}}$, where $[x]$ denotes the greatest integer less than or equal to $x$, is defined for all $x$ belonging to:

Options:

(A) all reals except integers

(B) all non-integers except the interval $[-1,1]$

(C) all integers except 0,-1,1

(D) all reals except the Interval $[-1,1]$

Sigma Percentile · Accepted Answer

### Analyzing the Numerator's Constraint

Let us first turn our attention to the numerator, $	ext{cosec}^{-1} x$. This is an inverse trigonometric function. We know that $	ext{cosec} 	heta = \frac{1}{\sin 	heta}$.

Because the sine function is bounded between $-1$ and $1$, its reciprocal, the cosecant function, must live outside this range. Specifically, $	ext{cosec} 	heta$ can never be between $-1$ and $1$.

Consequently, for the inverse function $	ext{cosec}^{-1} x$ to be defined, the input $x$ must satisfy $|x| \ge 1$. This translates to the interval $(-\infty, -1] \cup [1, \infty)$. These are the only regions where our numerator is allowed to exist.

### The Denominator's Trap

Now, let us look at the denominator: $\sqrt{x-[x]}$. We have two constraints here.

First, the expression inside the square root, $x-[x]$, must be greater than or equal to zero for the square root to yield a real number. Second, because this expression is in the denominator, it cannot be zero.

Combining these, we require:

$$x-[x] > 0$$

You might recognize $x-[x]$ as the definition of the fractional part of $x$, denoted as $\{x\}$. The fractional part of any real number is always in the interval $[0, 1)$.

The only time $\{x\}$ equals zero is when $x$ is an integer. Therefore, to satisfy $x-[x] > 0$, we must strictly forbid $x$ from being an integer.

### The Grand Intersection

We have our two conditions: $x \in (-\infty, -1] \cup [1, \infty)$ and $x 
otin \mathbb{Z}$. To find the domain of the entire function, we must find the intersection of these two sets.

Let us visualize this on the number line. We start with the regions $(-\infty, -1]$ and $[1, \infty)$. Now, we apply the filter: remove all integers.

The points $-1$ and $1$ are integers, so they must be removed, changing our closed intervals to open intervals: $(-\infty, -1)$ and $(1, \infty)$. Furthermore, we must remove all other integers like $-2, -3, 2, 3$, and so on.

Thus, the domain is the set of all real numbers in $(-\infty, -1) \cup (1, \infty)$ that are not integers. This is precisely described as:

**Domain** $= \{x \in \mathbb{R} : |x| > 1, x 
otin \mathbb{Z}\}$

Animated Solution for Mathematics - Inverse Trigonometric Functions: The real valued function $f (x) = \frac{cosec ^{- 1} x}{x - [ x ]}$ , where $[x]$ denotes the greatest integer less than or equal to $x$ , is defined for all $x$ belonging to:

Select Answer:

Visualized Solution

Function Structure Analysis

Numerator: Domain of $cosec^{- 1} x$

Visualizing Numerator Domain

Denominator: Square Root Condition

Fractional Part Property

Excluding Integers

Visualizing Denominator Domain

The Intersection

Final Domain Visualization

Final Conclusion

The Sigma Insight: Domain and Range of Inverse Trigonometric Functions

Analyzing the Numerator's Constraint

The Denominator's Trap

The Grand Intersection

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