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

Verified step by step guidance

1. First, identify the function given: \( f(x) = \lfloor x \rfloor \), where \( \lfloor x \rfloor \) denotes the floor function, which means the greatest integer less than or equal to \( x \).
2. Next, substitute the given expression for \( x \) into the function. The problem states \( x = x - (-\pi) \), which simplifies to \( x = x + \pi \).
3. Since the problem asks to find the value of the function for this \( x \), rewrite the function as \( f(x) = \lfloor x + \pi \rfloor \).
4. To evaluate \( f(x) \), you need a specific numerical value for \( x \). If \( x \) is given or assumed, add \( \pi \) to that value.
5. Finally, apply the floor function to the result of \( x + \pi \) by finding the greatest integer less than or equal to \( x + \pi \). This will give you \( f(x) \).

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 behavior.

Function Evaluation

Function evaluation involves substituting a given value of x into the function's expression and simplifying to find the output. This process requires careful handling of the input, especially when the function includes operations like subtraction or special functions.

Handling Expressions with π

When expressions involve π, it is important to recognize its approximate value (about 3.14159) and how to manipulate it algebraically. This helps in simplifying expressions like x - (-π) to x + π before applying the function.