Find the value of the function for the given value of x. See Example 3. ƒ(x)=-[[-x]], for x=2.5

Verified step by step guidance

1. Identify the function and the value of x given in the problem. The function is ƒ(x) = -[[-x]] and the value of x is 2.5.
2. Substitute the value of x into the function. Replace x with 2.5 in the function, so it becomes ƒ(2.5) = -[[-2.5]].
3. Evaluate the expression inside the inner brackets first. Calculate -2.5, which is the negation of the value of x.
4. Apply the floor function, denoted by the brackets [ ], to the result from the previous step. The floor function gives the greatest integer less than or equal to the number inside the brackets.
5. Finally, apply the negative sign outside the brackets to the result of the floor function to get the value of the function at x = 2.5.

Key Concepts

Function Notation

Function notation, such as f(x), represents a relationship where each input x corresponds to exactly one output. Understanding this notation is crucial for evaluating functions, as it indicates how to compute the output based on the given input value.

Greatest Integer Function

The greatest integer function, denoted as [[x]], returns the largest integer less than or equal to x. For example, [[2.5]] equals 2. This concept is essential for solving the problem, as it directly affects the output of the function f(x) when evaluating it at a non-integer input.

Negative Sign in Functions

The negative sign in front of a function, such as -[[x]], indicates that the output will be the opposite of the value produced by the function inside the brackets. This is important for correctly determining the final value of f(x) after applying the greatest integer function to the input.