Range of sin inverse [x^2+1/2] + cos inverse [ x^2-1/2] is where [] denotes greatest integer function

To find the range of the expression sin inverse [x² + 1/2] + cos inverse [x² - 1/2], where the brackets denote the greatest integer function, we need to first understand the individual ranges of the sine inverse and cosine inverse functions, as well as how they interact with the greatest integer function.

Understanding the Components

The sine inverse function, sin⁻¹(x), has a range of [-π/2, π/2], and the cosine inverse function, cos⁻¹(x), ranges from [0, π]. However, these ranges apply when dealing with the actual values of x, not necessarily the greatest integer function applied to x² + 1/2 and x² - 1/2.

Analyzing the Expressions

Let's break down each term:

• For sin⁻¹([x² + 1/2]): The term x² + 1/2 will always yield values greater than or equal to 1/2 since x² is always non-negative. The greatest integer function [x² + 1/2] will take integer values starting from 1 and can go as high as necessary depending on the value of x.
• For cos⁻¹([x² - 1/2]): Similarly, x² - 1/2 can take on a range of values. The expression is non-negative when x² is greater than or equal to 1/2, which occurs when |x| ≥ 1/√2. The greatest integer function [x² - 1/2] will yield integer values starting from 0 when x = ±1/√2 or higher.

Finding the Range of Each Function

Given these insights, we can analyze the possible integer values:

• For sin⁻¹, [x² + 1/2] will take integer values starting from 1 (for x = 0) and can increase indefinitely as x increases.
• For cos⁻¹, [x² - 1/2] will start at 0 (when x is ±1/√2) and can also increase indefinitely.

Combining the Results

Next, we need to add the ranges of these two functions together:

• When [x² + 1/2] = n (where n is an integer ≥ 1), sin⁻¹(n) produces values in the range of sin⁻¹(n) which will increase as n increases.
• When [x² - 1/2] = m (where m is an integer ≥ 0), cos⁻¹(m) produces values ranging from cos⁻¹(0) = π/2 to cos⁻¹(m) which decreases as m increases.

The sum sin⁻¹(n) + cos⁻¹(m) will thus yield a combination of values that can be assessed based on the integers chosen for n and m.

Final Thoughts on the Range

Ultimately, the range of sin⁻¹([x² + 1/2]) + cos⁻¹([x² - 1/2]) will depend on the integers n and m. As n and m can take on larger values, the sum can vary widely. The lowest possible value occurs when both are at their minimum, leading to a minimum sum of sin⁻¹(1) + cos⁻¹(0) = π/2. As n and m increase, the values of sin⁻¹(n) and cos⁻¹(m) will continue to expand the range. Thus, the overall range can be expressed as [π/2, ∞) since sin⁻¹(n) can approach π/2 and cos⁻¹(m) can contribute values that expand indefinitely.