Find the value of the function for the given value of x. See Example 3. ƒ(x)=[[x]], for x=-√2

Verified step by step guidance

1. Understand the function notation: here, ƒ(x) = [[x]] represents the greatest integer function (also known as the floor function), which returns 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 x is approximately -1.414.
3. Determine the greatest integer less than or equal to -1.414. This means finding the largest integer that is not greater than -1.414.
4. Since -1 is greater than -1.414, it cannot be the floor. The next integer less than -1.414 is -2, which satisfies the condition of being less than or equal to x.
5. Therefore, the value of the function ƒ(-\sqrt{2}) is the integer you found in the previous step.

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' a real number to the nearest integer.

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.

Properties of Square Roots and Negative Numbers

The square root of a positive number is always non-negative, but the input here is -√2, a negative number. Understanding how to handle negative inputs and their position on the number line is essential for applying the floor function correctly.