Find the value of the function for the given value of x. ƒ(x)=[[x]], for x=-√2

Verified step by step guidance

1. Understand the function notation: here, ƒ(x) = [[x]] represents the floor function, which means ƒ(x) gives the greatest integer less than or equal to x.
2. Identify the given value of x, which is x = -\sqrt{2}. Recall that \sqrt{2} is approximately 1.414, so -\sqrt{2} is approximately -1.414.
3. Apply the floor function to x = -\sqrt{2}. This means we need to find the greatest integer less than or equal to -1.414.
4. Determine the integer values around -1.414: the integers less than or equal to -1.414 are ..., -3, -2, -1, ... Among these, the greatest integer less than or equal to -1.414 is -2.
5. Conclude that ƒ(-\sqrt{2}) = -2, since -2 is the greatest integer less than or equal to -1.414.

Key Concepts

Floor Function (Greatest Integer Function)

The floor function, denoted by [[x]], returns the greatest integer less than or equal to x. For example, [[3.7]] = 3 and [[-1.2]] = -2. It essentially 'rounds down' any real number to the nearest integer below or equal to it.

Evaluating Functions at Specific Values

To find the value of a function at a given input, substitute the input value into the function's expression and simplify. This process helps determine the output corresponding to the input, which is essential for understanding function behavior.

Properties of Square Roots and Negative Numbers

The square root of a positive number is always non-negative, but when evaluating functions, the input can be negative, such as -√2. Understanding how to handle negative inputs and irrational numbers like √2 is important for accurate function evaluation.