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Chapter 1: Problem 51

In Exercises 51-56, sketch the graph of the function. \(g(x) = - [[x]]\)

Short Answer

Expert verified
The graph is a stair-shaped curve, with a negative slope. It jumps down at every integer value of \(x\).

Step by step solution

01

Understanding the Floor Function

Identify the output of the floor function by giving it some values. The function \([[x]]\) will output the largest whole number less than or equal to \(x\). This means, for any given integer \(x\), the function will output \(x\). However, for any non-integer value \(x\), it will round down to the nearest integer. For example, \([[1.2]] = 1\), \([[5]] = 5\), and \([[3.8]] = 3\).
02

Applying the Negation

Since the function is \(g(x) = - [[x]]\), the outputs from \([[x]]\) will be negated. This means the output value will be the negative of the value from step 1. For example, if \(x = 1.2\), then \(g(x) = - [[1.2]] = -1\). If \(x = 5\), then \(g(x) = - [[5]] = -5\). And if \(x = 3.8\), then \(g(x) = - [[3.8]] = -3\).
03

Sketching the Graph

Start plotting some points on a graph. Recognize the pattern of step lines between each integer \(x\)-value. At each integer, the function jumps down to the next lower value. The graph is a series of steps that decrease in value at every integer value of \(x\). The plotted points create a negatively sloped stair-shaped curve.

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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 is a special function that rounds a number down to the nearest whole number. This means that regardless of the decimal part of a given number, it takes the largest integer that is less than or equal to the original number.
For example:
  • For 3.7, the floor function \([3.7]\) equals 3, since 3 is the largest integer less than or equal to 3.7.
  • For -2.3, the floor function \([-2.3]\) equals -3, as the nearest lower integer is -3.
Understanding this behavior helps when graphing functions that involve rounding, as each non-integer value maps to the same integer outcome.
step graph
A step graph is a visual representation of a function like the floor function. It appears as a series of horizontal line segments, giving it a characteristic stair-step appearance.
Here's what you need to know:
  • Each step represents a range of input values that all produce the same output value. This is due to the "jump" at each integer value.
  • In the case of the graph for the function \(g(x) = - [[x]]\), each step begins just after an integer and continues until the next integer, where a sudden drop or step to the next line segment occurs.
When sketching a step graph, you'll create horizontal lines that drop at each whole number, resulting in a staircase shape that descends along the x-axis.
integer function
An integer function simplifies numbers by converting them into integers. It's like reducing a complex form into something simple, using integers.
Consider these points:
  • The floor function is a type of integer function because it outputs integers by taking the nearest lower integer from any real number.
  • Doing this repeatedly across numbers forms a sequence of integers, each "step" mapping a range of real numbers to a specific integer output.
Integer functions often display non-continuous graphs. This non-continuity is visible in the breaks or jumps at each integer point, making them useful in real-life scenarios where measurements must be whole numbers.

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Most popular questions from this chapter

In Exercises 63-76, determine whether the function has an inverse function. If it does, find the inverse function. \(f(x) = \frac{1}{x^2}\)

The direct variation model \(y = kr^n\) can be described as "\(y\) varies directly as the \(n\)th power of \(x\)," or "\(y\) is ________ ________ to the \(n\)th power of \(x\)."

CAPSTONE Describe and correct the error. Given \(f(x) = \sqrt{x-6}\), then \(f^{-1} (x) = \frac{1}{\sqrt{x-6}}\).

TAXES Property tax is based on the assessed value of a property. A house that has an assessed value of \(\$150,000\) has a property tax of \(\$5520\). Find a mathematical model that gives the amount of property tax \(y\) in terms of the assessed value \(x\) of the property. Use the model to find the property tax on a house that has an assessed value of \(\$225,000\).

In Exercises 25-54, \(g\) is related to one of the parent functions described in Section 1.6. (a) Identify the parent function \(f\). (b) Describe the sequence of transformations from \(f\) to \(g\). (c) Sketch the graph of \(g\). (d) Use function notation to write \(g\) in terms of \(f\). $$ g(x)=2(x-7)^{2} $$
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