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Chapter 2: Problem 78

Evaluate the step function defined by \(f(x)=[x]\) for the given values of \(x\). $$ f(6) $$

Short Answer

Expert verified
f(6) = 6

Step by step solution

01

Understand the Step Function

The step function, also known as the floor function, denoted by \( [x] \), returns the greatest integer less than or equal to \( x \). For any real number input \( x \), \( [x] \) is the largest integer such that \( [x] \) 鈮� \( x \).
02

Substitute the Given Value

Substitute \( x = 6 \) into the step function \( f(x) = [x] \).
03

Evaluate the Step Function

Since \( 6 \) is an integer, \( [6] \) is simply \( 6 \). Hence, \( f(6) = 6 \).

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Key Concepts

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

floor function
The floor function, often seen in math as \( \lfloor x \rfloor \), is an operation that takes a real number and maps it to the greatest integer less than or equal to that number. For example, \( \lfloor 4.7 \rfloor = 4 \). It's like rounding down to the nearest whole number.
This function comes in handy when dealing with numbers that are not whole since it helps to simplify and work with integers.
Notice that the floor function keeps any number that is already an integer untouched. For instance, \( \lfloor 5 \rfloor = 5 \).
But if you have a negative number, \( \lfloor -2.3 \rfloor \) would give you \( -3 \), since \( -3 \) is the greatest integer less than or equal to \( -2.3 \).
greatest integer function
You might have heard the floor function referred to as the 'greatest integer function.' This is because it finds the greatest integer that is not more than the given input.
For example:
  • For \( 3.8 \), the greatest integer not exceeding \( 3.8 \) is \( 3 \), so \( \lfloor 3.8 \rfloor = 3 \).
  • For \( -1.2 \), the greatest integer not exceeding \( -1.2 \) is \( -2 \), thus \( \lfloor -1.2 \rfloor = -2 \).
This understanding is critical in scenarios where rounding down has practical use, like in computations involving discrete values or in scenarios requiring exact integer counts.
evaluating step functions
Evaluating step functions involves applying the floor function principles. When given a step function of the form \( f(x) = \lfloor x \rfloor \), you're essentially determining the floor value of \( x \).
For instance, let's look at the step function given in the original problem, \( f(x) = \lfloor x \rfloor \), and evaluate it for specific values of \( x \):
  • If \( x = 4.5 \), then \( f(4.5) = \lfloor 4.5 \rfloor = 4 \).
  • If \( x = -0.8 \), then \( f(-0.8) = \lfloor -0.8 \rfloor = -1 \).
When dealing with integer inputs, the function returns the same integer, making it straightforward.
integer inputs in step functions
The fascinating thing about step functions with integer inputs is their predictability. When you input an integer into a step function like \( f(x) = \lfloor x \rfloor \), it always returns the input value itself.
This property simplifies evaluations significantly. For example:
  • Evaluating for \( x = 6 \), we find \( f(6) = \lfloor 6 \rfloor = 6 \).
  • If \( x = -3 \), then \( f(-3) = \lfloor -3 \rfloor = -3 \).
This consistency allows step functions to be a reliable tool in mathematics, providing stable outcomes when dealing with whole numbers.

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