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Chapter 1: Problem 33

Graphing a step Function. Sketch the graph of the function. $$g(x)=[ | x]-1$$

Short Answer

Expert verified
The graph of \(g(x) = [|x|] - 1\) is a combination of step and constant functions. It is constant at y = -1 for \(x < 0\) and a series of steps decreasing by 1 at each positive integer x for \(x \geq 0\).

Step by step solution

01

Understanding Absolute Function

The absolute value function \(|x|\), is actually composed of two linear functions. For \(x < 0\), \(|x| = -x\) and for \(x \geq 0\), \(|x| = x\). This function will mirror inputs from the negative x-axis onto positive y-axis.
02

Understanding Floor Function

The floor function, represented as [], returns the greatest integer less than or equal to x. For example, [2.1] = 2 and [-2.1] = -3. The graph has step like discontinuities at each integer.
03

Combined Absolute and Floor Function

Now, consider the combined function [|x|]. Here the absolute function works first then the floor function applies. So, for \(x \geq 0\), the output is the integer part of x (i.e., x itself if x is an integer), and for \(x < 0\), the output is 0 because it's the greatest integer less than |x| for \(x < 0\).
04

Graphing g(x) = [|x|] - 1

The given function is a vertical shift of the combined function down by 1 unit. For \(x \geq 0\), the graph is a step function decreasing by 1 unit at each positive non-integer x and at x = 0, g(x) = -1. For \(x < 0\), the graph is a constant function at y = -1 because [|x|] is always 0 for \(x < 0\). Create this as a graph on the x-y axes to exhibit this behavior visually.

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Key Concepts

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

Exploring the Absolute Value Function
The absolute value function, represented as \(|x|\), is a fascinating mathematical construct that you'll frequently encounter in mathematics. The core idea is to determine a number's distance from zero, which is always non-negative. It's like taking any number and simply erasing its sign, because we're only interested in how far it is from the origin on a number line.
\
\Here's the breakdown:
    \
  • For any positive number or zero, \(|x|\) is just itself, \(|x| = x\).
  • For a negative number, the absolute value negates it to make it positive, \(|x| = -x\).
    • \
This dual-definition might seem tricky, but just remember that \(|x|\) always keeps the value positive. When you visualize it on a graph, it reflects the negative part of a graph upward along the x-axis.
Understanding the Floor Function
The floor function is like the mathematical equivalent of rounding down. It's often depicted with the notation \([x]\), not to be confused with parentheses. What it does is quite simple: it takes any real number and "floors" it to the nearest lower integer.
\
\Think of it as finding a number's "home" on the integer scale to the left of it. Here are examples to clarify:
    \
  • If you take the input 2.9, \([2.9] = 2\). It's been rounded down from 2.9 to 2.
    • \
  • Should you have -1.2 as input, \([-1.2] = -2\). It effectively rounds from left on number line, which numerically goes further from zero.

\The floor function typically creates a staircase-like graph where values jump at integer boundaries.
Graphing Piecewise Functions
Piecewise functions combine different rules or equations over various parts of their domain. In the case of our function, \([|x|] - 1\), we're blending together the absolute and floor functions, hence creating a step function.
\
\When graphing piecewise functions, it's efficient to:
    \
  • Identify each segment of the piecewise function. For instance, \([|x|]\) will affect \([x]\) based on whether \(|x|\) is zero or positive.
    • \
  • Define clear boundaries. In this case, reduction of 1 moves the floor step pattern vertically, affecting each integer section on the graph.

\Piecewise functions should be graphed section by section, evaluating how the blended rules manifest on the x-y plane.
The Effects of Vertical Shift
Vertical shifts in graphs change the entire graph's position along the y-axis. In simpler words, such a transformation either raises or lowers the graph as a whole. It's simple yet crucial, as it can significantly alter a graph's behavior.
\
\In the function \([|x|] - 1\), the subtraction of 1 indicates a downward vertical shift by 1 unit.
    \
  • For \([|x|]\), the lowest value, at \(x < 0\), changes from 0 to -1 after the shift, reflecting this on the y-axis.
    • \
  • For \((x \geq 0)\), each integer step decreases by 1 as you move across the positive x-axis.

\Think of it like lowering a staircase by one step; each step remains identical, but the entire staircase sits 1 unit lower than before. This shift is visually evident when you plot the graph.

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Most popular questions from this chapter

Mathematical models that involve both direct and inverse variation are said to have _________ variation.

Comparing Slopes Use a graphing utility to compare the slopes of the lines \(y=m x\) , where \(m=0.5,1,2,\) and \(4 .\) Which line rises most quickly?Now, let \(m=-0.5,-1,-2,\) and \(-4 .\) Which line falls most quickly? Use a square setting to obtain a true geometric perspective. What can you conclude about the slope and the "rate" at which the line rises or falls?

Matching and Determining Constants In Exercises \(85-88\) , match the data with one of the following functions $$ f(x)=c x, g(x)=c x^{2}, h(x)=c \sqrt{|x|}, \text { and } r(x)=\frac{c}{x} $$ and determine the value of the constant \(c\) that will make the function fit the data in the table. $$ \begin{array}{|c|c|c|c|c|c|}\hline x & {-4} & {-1} & {0} & {1} & {4} \\\ \hline y & {-32} & {-2} & {0} & {-2} & {-32} \\ \hline\end{array} $$

Finding a Relationship for Equidistance, find a relationship between \(x\) and \(y\) such that \((x, y)\) is equidistant (the same distance) from the two points. $$(4,-1),(-2,3)$$

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