Problem 30 - Thomas Calculus

Question

Graph \(y=\sin x\) and \(y=\lceil\sin x\rceil\) together. What are the domain and range of \(\lceil\sin x\rceil\) ?

Step-by-Step Solution

Answer

Domain is all real numbers; range is {-1, 0, 1}.

Step 1: Understand the Function

The function given is the ceiling of the sine function, denoted as \(y=\lceil\sin x\rceil\). The sine function \(y=\sin x\) oscillates between -1 and 1 as \(x\) varies over the real numbers, repeating its values every \(2\pi\) units.

Step 2: Graph the Basic Sine Function

Graph \(y = \sin x\) over one period from \(x = 0\) to \(x = 2\pi\). It starts at 0, rises to 1 at \(x = \frac{\pi}{2}\), decreases to 0 at \(x = \pi\), drops to -1 at \(x = \frac{3\pi}{2}\), and returns to 0 at \(x = 2\pi\).

Step 3: Apply Ceiling to the Sine Function

The ceiling function \(\lceil x \rceil\) rounds its input to the nearest integer greater than or equal to \(x\). Apply this to \(y=\sin x\). When \(\sin x\) is between -1 and 0, \(\lceil\sin x\rceil = 0\), and when \(\sin x\) is between 0 and 1, \(\lceil\sin x\rceil = 1\). At points where \(\sin x = 0\) or \(\sin x = 1\), the value of \(\lceil \sin x \rceil\) is 0 and 1, respectively.

Step 4: Observe Properties for Domain and Range

The domain of \(\lceil\sin x\rceil\) is all real numbers since \(\sin x\) is defined for all real \(x\). The range of \(\lceil\sin x\rceil\) over any full period is the set of integers \{-1, 0, 1\} because the output is set by the ceiling of possible \(\sin x\) values which are -1 to 1.

Key Concepts

Ceiling Function

The ceiling function, denoted as \(\lceil x \rceil\), is a mathematical concept that rounds a number up to the nearest integer. This means it takes any real number and moves it up to the smallest integer that is greater than or equal to the input number. A few things to note about the ceiling function are:

• If the number is already an integer, the ceiling is the number itself.
• If the number is positive and not an integer, the ceiling rounds it up to the next whole number.
• If the number is negative and not an integer, the ceiling function rounds it towards zero, often making it closer to zero.

In the context of sinusoidal functions, applying the ceiling function to \(y = \sin x\) means that for any given value of \(x\), the resultant \(y\) value will be the smallest integer greater than or equal to \(\sin x\). Hence, this modification to the sine function results in an output that takes on integer values like 0 or 1, depending on whether \(\sin x\) is positive, zero, or negative.

Sine Function

The sine function, represented as \(y = \sin x\), is one of the fundamental trigonometric functions. It describes a repetitive wave pattern that oscillates between values of -1 and 1. This periodic nature makes the sine function very important in various fields of science and engineering.

• **Periodicity:** The sine function repeats every \(2\pi\) radians. This means exactly the same values are generated every \(2\pi\).
• **Amplitude:** The sine wave has an amplitude of 1, meaning it reaches a maximum of 1 and a minimum of -1.
• **Key Points:** Critical points of the sine wave include where it crosses the x-axis (0, \(\pi\), and \(2\pi\)), its peaks (\(\frac{\pi}{2}\)) and troughs (\(\frac{3\pi}{2}\)).
• **Phase Shift:** The function may shift horizontally with phase changes but generally follows the same pattern.

This efficient, predictable behavior of the sine wave is a cornerstone in understanding more complex oscillating processes.

Domain and Range

In mathematical terms, the domain is the set of all possible inputs (x-values) for which a function is defined, while the range is the set of all possible outputs (y-values) the function can produce.

Domain of \(\lceil\sin x\rceil\):
Since the sine function \(\sin x\) is defined for all real numbers, the ceiling of the sine, \(\lceil \sin x \rceil\), shares the same domain, which encompasses all real numbers, \((-\infty, \infty)\). The sine function doesn't have any breaks or undefined spots, making its ceiling also seamless across the number line.

Range of \(\lceil\sin x\rceil\):
The range of \(\lceil \sin x \rceil\) is dictated by the ceiling effect applied to \(\sin x\), which outputs values between -1 and 1. Consequently, \(\lceil \sin x \rceil\) will only yield integer values, specifically \{-1, 0, +1\}.

This means for any given \(x\), the possible output after applying the ceiling function to \(\sin x\) is limited to these integers, transforming the smooth wave of the sine function into discrete "steps".