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Chapter 2: Problem 79

Evaluate the step function defined by \(f(x)=[x]\) for the given values of \(x\). $$ f(-9) $$

Short Answer

Expert verified
f(-9) = -9

Step by step solution

01

Understand the function

The given function is a step function, specifically the floor function denoted by \( f(x) = [x] \). The floor function returns the greatest integer less than or equal to \( x \).
02

Identify the value of \(-9\)

Examine the given value of \( x \), which is \(-9\). Since \(-9\) is already an integer, the floor function will return the integer itself.
03

Apply the floor function

Apply the floor function to \(-9\): \[ f(-9) = [-9] = -9 \]

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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, also known as the greatest integer function, maps any real number to the largest integer less than or equal to it. In mathematical notation, it's written as \( f(x) = [x] \). This function is particularly useful in problems where you need to round down to the nearest whole number. For instance, \( [3.7] \) will yield 3, as 3 is the largest integer less than 3.7.

This function can handle negative numbers too. For example, if \( x = -2.4 \), \( [x] = -3 \), since -3 is the greatest integer less than -2.4. Understanding this concept is crucial for solving step function problems efficiently.
greatest integer function
The greatest integer function is essentially another name for the floor function. Both denote the same concept: mapping any real number to the largest integer less than or equal to it. However, do not confuse it with the

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Most popular questions from this chapter

A sales person makes a base salary of \(\$ 2000\) per month. Once he reaches \(\$ 40,000\) in total sales, he earns an additional \(5 \%\) commission on the amount in sales over \(\$ 40,000 .\) Write a piecewise-defined function to model the sales person's total monthly salary \(S(x)\) (in \(\$)\) as a function of the amount in sales \(x\).

Determine if the function is even, odd, or neither. $$ q(x)=\sqrt{16+x^{2}} $$

Refer to the functions \(f\) and \(g\) and evaluate the functions for the given values of \(x\). \(f=\\{(2,4),(6,-1),(4,-2),(0,3),(-1,6)\\} \quad\) and \(\quad g=\\{(4,3),(0,6),(5,7),(6,0)\\}\) $$(f \circ f)(-1)$$

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