Find the value of the function for the given value of x. See Example 3. ƒ(x)=[[x]], for x=-√2
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Identify the function and the input value. The function given is ƒ(x)=[[x]], and the input value is x=-√2.
Understand the notation [[x]], which represents the greatest integer less than or equal to x, also known as the floor function.
Calculate the value of -√2. Since the square root of 2 is approximately 1.414, -√2 is approximately -1.414.
Apply the floor function to -1.414. The greatest integer less than -1.414 is -2.
Thus, ƒ(-√2) is equal to the floor of -1.414, which is -2.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Function Notation
Function notation, such as ƒ(x), represents a mathematical 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.
The greatest integer function, denoted as [[x]], returns the largest integer less than or equal to x. This concept is essential for solving the problem, as it requires determining the integer part of the input value, which can significantly affect the output of the function.
Evaluating a function involves substituting a specific value into the function's expression to find the corresponding output. In this case, substituting x = -√2 into the greatest integer function requires understanding how to handle irrational numbers and their relationship to integers.