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

Verified step by step guidance

1. First, understand the function notation given: ƒ(x) = [[x]]. The double brackets [[x]] typically represent the greatest integer function (also known as the floor function), which means ƒ(x) returns the greatest integer less than or equal to x.
2. Next, identify the value of x given in the problem: x = x - (-\pi). Simplify this expression by recognizing that subtracting a negative is the same as adding, so x = x + \pi.
3. Since the function is ƒ(x) = [[x]], substitute the simplified value of x into the function: ƒ(x) = [[x + \pi]].
4. To find the value of ƒ(x), evaluate the expression inside the greatest integer function, which is x + \pi. This means you add \pi (approximately 3.14159) to the given x value.
5. Finally, apply the greatest integer function to the result of x + \pi by finding the greatest integer less than or equal to that sum. This will give you the value of ƒ(x).

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. Understanding how to evaluate this function is essential for solving problems involving piecewise or stepwise outputs.

Function Evaluation

Function evaluation involves substituting a given value of x into the function's formula and simplifying to find the output. This requires careful handling of expressions inside the function before applying any operations like the floor function.

Simplifying Expressions Involving π

When expressions include π, such as x - (-π), it is important to correctly simplify by recognizing that subtracting a negative is equivalent to addition. This ensures accurate substitution and evaluation of the function.