Domain Of A Function

We are given the function and need to find its domain.

There are two conditions that must be satisfied for the function to be defined:

1. The expression inside the square root must be non-negative:
2. The argument of the inverse cosine function must be between -1 and 1 (inclusive):

Since is always non-negative, we can rewrite the second condition as:

Squaring all parts, we get:

Adding 1 to all parts, we get:

Combining both conditions and , we find that the domain is .

Thus, the correct option is (A) .

The given function is , where is the greatest integer less than or equal to x.

For the function to be defined, the expression inside the square root must be positive, i.e., .

Let . Then, we have .

Factoring the quadratic expression, we get .

This inequality holds when or .

So, we have two cases:

Case 1: . Since is an integer, this means . Thus, . Therefore, .

Case 2: . Since is an integer, this means . Thus, . Therefore, .

Combining the two cases, the domain of the function is .

The function is defined for all except for (since the denominator cannot be zero). Now, let's analyze whether the function is injective (one-to-one) and surjective (onto).
Injective (One-to-one): A function is injective if, for every pair of distinct elements and , . To check if is injective, we solve for and such that: Cross-multiply and solve: Simplifying: Therefore, the function is injective. **Surjective (Onto)**: A function is surjective if for every element , there exists an such that . Let's examine if the function covers all real numbers: We can see that there are values of for which the function will not reach certain values in . For example, the function cannot take the value since: This results in a contradiction, meaning the function is not surjective.

Therefore, is injective but not surjective.

The function is defined as: In the II quadrant (where ):

• is positive
• is negative

Hence, ⇒ As increases from to , increases (less negative to 0), so becomes less negative (i.e., increases from to 0).

Correct Answer: increases from - to 0