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Chapter 1: Problem 111

Think About It Describe how the functions $$f(x)=3+[x]\( and \)g(x)=3-[1-x]$$ differ.

Short Answer

Expert verified
The function \( f(x) = 3 + [x] \) is a basic upward shifted floor function. The function \( g(x) = 3 - [1-x] \) is a reflection of the floor function, flipped about the y-axis and shifted down. These modifications induce different values at the same input depending on the function.

Step by step solution

01

Analyze the function \( f(x)=3+[x] \)

The function \( f \) is a floor function \( [x] \) which is then added to a constant \( 3 \). The base floor function \([x]\) returns the greatest integer less than or equal to \( x \). Additions on the outside of the floor function do not change the behaviour of the floor function, they simply shift it up on the y-axis. Therefore, each point on the function \([x]\) is simply shifted 3 units upwards to make the final function \( 3+ [x] \).
02

Analyze the function \( g(x)=3-[1-x] \)

The function \( g(x) \) first subtracts \( x \) from \( 1 \), applies the floor function to the result and then subtracts this from 3. A subtraction inside the floor function changes the function's behaviour. It means that for each value \( x \), we first 'flip' it by subtracting it from \( 1 \), then we take the floor. The inner operation creates a reflection of the base floor function across the y-axis. The outer operation, subtracting from 3, then shifts the graph 3 units down on the y-axis. Therefore, the function \( g(x) \) is a flipped version of the floor function, shifted 3 units down.
03

Compare the two functions

The function \( f(x) = 3 + [x] \) is a regular floor function, shifted up 3 units, whereas \( g(x)=3-[1-x] \) is a reflected, upside-down version of the floor function, also shifted down 3 units. These differences are manifested in the behaviours of the two functions. \( f(x) \) will round any \( x \) down to the nearest lower integer and then add 3. Conversely, \( g(x) \) will flip \( x \) by subtracting it from 1, round down to the nearest integer, and then subtract that from 3, which can result in different values even for the same \( x \).

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Key Concepts

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

Greatest Integer
The greatest integer function, often represented by the floor function \([x]\), is an important concept in algebra and calculus. This function finds the largest integer less than or equal to a given number.
  • For example, \([3.7] = 3\) because 3 is the greatest integer less than 3.7.
  • Similarly, \([-2.3] = -3\) as -3 is the largest whole number less than -2.3.
Understanding this function helps in various mathematical processes, such as simplifying functions for closer examination.When we deal with the expression \(f(x) = 3 + [x]\), you're working with the floor function added to a constant, shifting all its values by 3 units upwards on the vertical axis.This is a linear step function, with each interval maintaining a constant output until the next integer threshold is crossed.
Function Analysis
Function analysis delves into dissecting and understanding different mathematical functions, their behavior, and interactions with variables. Let's break down step-by-step how to analyze the given functions.**Function \(f(x) = 3 + [x]\):**- Begin by recognizing the floor function \([x]\).- Understand that adding 3 shifts this entire function upwards by 3 units.**Function \(g(x) = 3 - [1 - x]\):**- Subtracting \(x\) from 1 inside the floor function \([1-x]\) essentially flips the value around the origin before applying the floor function.- The operation \(3 - [1-x]\) shifts the results down by 3 units after flipping.Through analysis, we see that while \(f(x)\) moves vertically up by constant steps, \(g(x)\) not only flips left-right but also steps down, appearing quite different when graphed.
Graph Transformation
Graph transformation involves altering the position or shape of a graph through various operations. The two functions, \(f(x) = 3 + [x]\) and \(g(x) = 3 - [1 - x]\), demonstrate important aspects of these transformations.When you add a constant to a function, like how \(f(x)\) adds 3 to the floor function, the entire graph shifts upwards.- This does not alter the horizontal line intervals of the floor function itself.For \(g(x) = 3 - [1 - x]\):
  • First, the argument \([1-x]\) flips the graph horizontally. Imagine folding the function line over the y-axis.
  • Then, by subtracting from 3, the graph adjusts downward by 3 units.
In graph transformations, each modification鈥攚hether a shift, flip, or stretch鈥攁ffects the visual appearance of the graph. This highlights the powers of graph transformations in visualizing function behaviors.

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