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Chapter 2: Problem 36

If \(f(x)=|x|+[x],\) the value of \(f(-2.5)+f(1.5)\) is (A) \(-2\) (B) 1 (C) 1.5 (D) 2 (E) 3

Short Answer

Expert verified
The value of \(f(-2.5)+f(1.5)\) is 2

Step by step solution

01

Computing f(-2.5)

Firstly calculate the designated function for -2.5. The absolute value of -2.5 is 2.5 and the floor value of -2.5 is -3. Therefore, \(f(-2.5) = |-2.5| + [-2.5] = 2.5 - 3 = -0.5.\)
02

Computing f(1.5)

Next, compute the function for 1.5. The absolute value of 1.5 is 1.5 and the floor value of 1.5 is 1. Therefore, \(f(1.5) = |1.5| + [1.5] = 1.5 + 1 = 2.5.\)
03

Adding the Values

Finally add the two values computed previously. Hence, \(f(-2.5)+f(1.5) = -0.5 + 2.5 = 2\). Thus, the solution is 2.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Absolute Value Function
The absolute value of a number is its distance from zero on the number line, regardless of direction. This means that whether the number is positive or negative, the absolute value will always be non-negative. You can think of it as a number's "magnitude" without regard to its sign.

  • If you take a positive number, its absolute value is the number itself, for example, \(|3| = 3\).
  • If you take a negative number, the absolute value is its positive counterpart, as in \(|-3| = 3\).
  • The absolute value of zero is zero, \(|0| = 0\).
Understanding absolute values is key, particularly when dealing with expressions that include both negative and positive numbers, as it focuses solely on the size, not the sign, of the number.
Floor Function
The floor function, often denoted by \([]\), is used to find the greatest integer less than or equal to a given number. It essentially "rounds down" a number to the nearest integer.
  • If the number is already an integer, the floor function returns that number, such as \[ [3] = 3 \].
  • If you have a positive decimal, it rounds down towards zero, for instance, \[ [2.7] = 2 \].
  • For negative numbers, it becomes a bit more interesting. For example, \[ [-2.5] = -3 \]. This is because \(-3\) is the largest integer less than \(-2.5\).
The floor function helps simplify and evaluate expressions by converting a continuous number into its integer form.
Function Evaluation
Function evaluation involves finding the output of a function for a particular input by following the rules given in the function's definition. It鈥檚 like a machine where you input a number, follow the process, and get an output.

Consider the function \(f(x) = |x| + [x]\). Here, for any input \(x\):
  • Calculate the absolute value \(|x|\), which provides a positive component.
  • Find the floor value \([x]\) to get an integer component.
  • Add these two components together to get the function value.
By taking specific inputs like \(-2.5\) and \(1.5\), you can see how these operations produce distinct outputs that, when added together, give the final result. This process helps us solve complex problems by systematically evaluating each part of the function.

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Most popular questions from this chapter

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