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

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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.

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 + \(\pi\).

Since the function is ƒ(x) = [[x]], substitute the simplified value of x into the function: ƒ(x) = [[x + \(\pi\)]].

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.

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).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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