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Chapter 2: Problem 80

Evaluate the step function defined by \(f(x)=[x]\) for the given values of \(x\). $$ f(-5) $$

Short Answer

Expert verified
f(-5) = -5

Step by step solution

01

Understand the Step Function

The step function is defined as \( f(x) = \lfloor x \rfloor \), which returns the greatest integer less than or equal to \( x \).
02

Identify the Input Value

In this problem, \( x \) is given as \(-5\).
03

Apply the Step Function

Evaluate the function using the floor definition: \( f(-5) = \lfloor -5 \rfloor \). Since \(-5\) is already an integer, \( \lfloor -5 \rfloor = -5 \).

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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 denoted as \(\floor{x}\), is a mathematical function that maps a real number to the largest integer less than or equal to that number. For example, \( \floor{3.7} = 3\) and \( \floor{-2.3} = -3\). This function is useful in various areas of math and computer science for simplifying and discretizing values.
Key properties of the floor function include:
  • \( \floor{x} \) is always less than or equal to \ x \.
  • \( \floor{x} \) is an integer.
  • If \( x\) is already an integer, then \( \floor{x} = x \).
This makes the floor function helpful when managing values that require rounding down to the nearest integer.
greatest integer function
The greatest integer function is another name for the floor function. This terminology is often used interchangeably but emphasizes its effect of finding the 'greatest' whole number that still satisfies the condition of being less than or equal to the input value. For example, \( f(-1.2) = \floor{-1.2} = -2 \).
Consider some examples:
  • If \( x = 4.9 \), then the greatest integer function gives \( f(x) = 4\).
  • If \( x = -3.4 \), then \( f(x) = -4 \).
  • If \( x = 7.0 \), then \( f(x) = 7 \) because 7 is already an integer.
The greatest integer function can be particularly practical when managing quantities that can only exist as whole numbers, like counting objects or people.
evaluating functions
When evaluating functions, especially step functions like the floor function, it鈥檚 crucial to understand the input and the function鈥檚 rule. Let's recap the exercise we analyzed earlier:
We have the function defined by \( f(x) = \floor{x} \) and need to evaluate \( f(-5) \). To solve this:
  • First, recognize that the input value is \-5 \.
  • Next, apply the step function by finding the greatest integer less than or equal to \-5 \.
  • Since \-5 is already an integer, the result is \ f(-5) = -5 \.
Breaking down the problem into these clear steps helps ensure accuracy and comprehension when evaluating functions, especially those that manipulate integer values.

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Most popular questions from this chapter

A car traveling \(60 \mathrm{mph}(88 \mathrm{ft} / \mathrm{sec})\) undergoes a constant deceleration until it comes to rest approximately 9.09 sec later. The distance \(d(t)\) (in ft) that the car travels \(t\) seconds after the brakes are applied is given by \(d(t)=-4.84 t^{2}+88 t,\) where \(0 \leq t \leq 9.09 .\) (See Example 5) a. Find the difference quotient \(\frac{d(t+h)-d(t)}{h}\). Use the difference quotient to determine the average rate of speed on the following intervals for \(t\) : b. [0,2]\(\quad(\) Hint \(: t=0\) and \(h=2)\) c. [2,4]\(\quad(\) Hint \(: t=2\) and \(h=2)\) d. [4,6]\(\quad(\) Hint \(: t=4\) and \(h=2)\) e. [6,8]\(\quad(\) Hint \(: t=6\) and \(h=2)\)

Graph the function. $$ k(x)=\operatorname{int}\left(\frac{1}{2} x\right) $$

From an equation in \(x\) and \(y\), explain how to determine whether the graph of the equation is symmetric with respect to the \(x\) -axis, \(y\) -axis, or origin.

Given \(f(x)=\frac{12}{x}\) a. Find the difference quotient (do not simplify). b. Evaluate the difference quotient for \(x=2,\) and the following values of \(h: h=0.1, h=0.01, h=0.001\), and \(h=0.0001\). Round to 4 decimal places. c. What value does the difference quotient seem to be approaching as \(h\) gets close to \(0 ?\)

Evaluate the step function defined by \(f(x)=[x]\) for the given values of \(x\). $$ f(-0.09) $$
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