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Chapter 2: Problem 48

Sketch the graph of \(f\) $$f(x)=\left\\{\begin{array}{ll} -1 & \text { if } x \text { is an integer } \\ -2 & \text { if } x \text { is not an integer } \end{array}\right.$$

Short Answer

Expert verified
Discrete points at \(y = -1\) for integers; a continuous line at \(y = -2\) for non-integers.

Step by step solution

01

Understand the Function

The function \( f(x) \) is a piecewise function: it takes the value \(-1\) if \(x\) is an integer and \(-2\) if \(x\) is not an integer. This behavior will influence drawing two different levels on the graph.
02

Identify Integer Values

Recognize that integer values are numbers like \(..., -3, -2, -1, 0, 1, 2, 3, ...\). For these values of \(x\), \(f(x) = -1\). Each plotted point for these integers on the graph should be marked at a level \(y = -1\).
03

Plot Integer Points

Plot points such as \((-3, -1)\), \((-2, -1)\), \((-1, -1)\), \((0, -1)\), \((1, -1)\), \((2, -1)\), \((3, -1)\), and so forth. These points should appear as dots on the graph line \(y = -1\).
04

Identify Non-Integer Values

Non-integer values include numbers such as \(-3.5, 2.1, \pi, \sqrt{2},\) etc. For these values of \(x\), the function assigns \(f(x) = -2\).
05

Plot Non-Integer Values

For non-integer values, the output is \(-2\), creating a densely packed line at \(y = -2\). This line will not contain gaps and should be smoothly continuous alongside the points plotted at \(y = -1\) for integers.
06

Connect Points and Lines

The graph includes a dotted line of discrete points at \(y = -1\) for integers and a solid continuous line at \(y = -2\) for all non-integers. Ensure the scattered nature of integer outputs and continuous nature of non-integer outputs is visually clear.

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Key Concepts

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

Graph Sketching
Graph sketching is a crucial skill that allows us to visualize functions and understand their behavior. For piecewise functions, the graph is often split into sections based on different conditions. Here, we're working with a function that changes based on whether the input is an integer or not.
  • For integer values, the function reaches a certain level.
  • For non-integers, it settles on a different level.
To sketch the graph of the piecewise function given, it's essential to distinguish these segments clearly.
The integer values form discrete points, easily spotted when plotted. Non-integer values, on the other hand, create a continuous line. Keeping these traits in mind makes sketching accurate and straightforward.
Integer Values
Integer values are whole numbers including negatives, zero, and positives, like \\(..., -3, -2, -1, 0, 1, 2, 3, ...\). They stand out as distinct positions on the graph due to their discrete nature. In our function:
  • Whenever \(x\) is an integer, \(f(x)\) assigns the value \ \(-1\).
  • These points appear only at the \(y = -1\) level on the graph.
To plot them:
  • Use points like \ \((-3, -1), (-2, -1), \ldots\).
  • These create a horizontal line of dots.
This line is visually distinct from the continuous sections, helping to identify integer inputs quickly.
Non-Integer Values
Non-integer values include any number that isn't a whole number. This category features fractions and irrationals like \\(-3.5, 0.2, \pi\), and step up beyond simple integers. For our function:
  • When \(x\) is a non-integer, \(f(x)\) yields \ \(-2\).
  • This is represented as a continuous line on \(y = -2\).
To illustrate this on our graph:
  • The non-integers should form an unbroken line across \(y = -2\).
  • It highlights the difference between distinct and continuous parts of the graph.
This continuous line shows the rich variety of numbers that fill the spaces between integers.
Step Function
A step function jumps from one level to another, similar to climbing stairs. Each "step" reflects a sudden change in value, not a smooth transition. In our scenario:
  • The function \(f(x)\) behaves like a step due to its integer and non-integer outputs.
  • For integers, the function is a "step" on the \(y = -1\) level.
  • For non-integers, it "steps" down to \(y = -2\).
The steps seen on the graph provide a clear method to understand how inputs are classified based on the type of number. This makes it evident that each input type has its dedicated outcome on the graph. Step functions play a significant role in representing systems with discrete levels or categories.

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Most popular questions from this chapter

Graph \(f\) in the viewing rectangle \([-12,12]\) by \([-8,8] .\) Use the graph of \(f\) to predict the graph of \(g .\) Verify your prediction by graphing \(g\) in the same viewing rectangle. $$f(x)=0.5 x^{2}-2 x-5 ; \quad g(x)=0.5 x^{2}+2 x-5$$

A certain paperback sells for 12 dollar. The author is paid royalties of \(10 \%\) on the first 10,000 dollar copies sold, \(12.5 \%\) on the next 5000 copies, and \(15 \%\) on any additional copies. Find a piecewise-defined function \(R\) that specifies the total royalties if \(x\) copies are sold.

Estimate the solutions of the inequality. $$|0.3 x|-2>2.2-0.63 x^{2}$$

Find two positive real numbers whose sum is 40 and whose product is a maximum.

An object is projected vertically upward from the top of a building with an initial velocity of \(144 \mathrm{ft} / \mathrm{sec} .\) Its distance \(s(t)\) in feet above the ground after \(t\) seconds is given by the equation $$ s(t)=-16 t^{2}+144 t+100 $$ (a) Find its maximum distance above the ground. (b) Find the height of the building.
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