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The sine function, denoted as \(y = \sin x\), is a fundamental aspect of trigonometry that is often represented visually as a smooth, wave-like curve. Its distinctive shape repeats every \(2\pi\) radians, which makes it a periodic function. This means after each interval of \(2\pi\), the curve begins its pattern again.
The sine curve starts at the origin point (0,0) on a graph. As you move along the x-axis, \(y = \sin x\) reaches its maximum value of 1 at \(x = \pi/2\). Then, it descends back to zero at \(x = \pi\), dips to its minimum value of -1 at \(x = 3\pi/2\), and returns to zero at \(x = 2\pi\). This sequence continues indefinitely in both the positive and negative x-axis directions, capturing the essence of its periodic nature.
Key characteristics of the sine function include:

• Ranges from -1 to 1, inclusive.
• Repeats every \(2\pi\).
• Is smooth and continuous.

The floor function, represented mathematically as \(\lfloor x \rfloor\), has a straightforward task: it rounds down a number to the nearest integer less than or equal to the given number. Imagine being on a staircase where you can only step down, never up.
This rounding process leads to a function characterized by sharp jumps from one integer to the next. As a result, the output of \(\lfloor x \rfloor\) is a piecewise constant function that produces a staircase-like graph. For example, \(\lfloor 3.7 \rfloor = 3\) and \(\lfloor -2.4 \rfloor = -3\).
When applied to the sine function, the floor function converts the smooth sine wave into a step-wise graph:the output is -1 when \(\sin x < 0\), 0 when \(0 \leq \sin x < 1\), and 1 precisely at \(\sin x = 1\). This creates a series of horizontal lines representing intervals of constant value.

The absolute value function, denoted as \(|x|\), takes any real number and transforms it into its non-negative equivalent. Essentially, it specifies the distance of a number from zero on the number line without considering direction.
Applying this to the sine function, \(|\sin x|\) transforms the wave so that all its negative values flip to become positive. The graph of \(|\sin x|\) will mirror the graph of \(\sin x\) for the positive intervals and reflect the negative intervals across the x-axis.

• Original negative values like -0.5 become 0.5.
• Zero remains unchanged as |0| = 0.

Through this transformation, the sine wave now consists entirely of non-negative values, showcasing the property that absolute values can only be zero or positive.

In mathematics, understanding the domain and range of a function helps us ascertain its behavior across various inputs and outputs.
The domain of a function refers to all possible input values (x-values) for which the function is defined. For \(\sin x\), and by extension \(|\sin x|\), the domain covers all real numbers because you can input any value of \(x\) and obtain an output. Thus, the domain is (-\infty, \infty).
In contrast, the range of a function refers to the possible output values (y-values) the function can produce. For \(|\sin x|\), the transformation caused by taking the absolute value ensures that all outputs lie between 0 and 1, inclusive. Therefore, its range is given by [0, 1], highlighting every value from 0 through 1 is possible, even if \(x\) varies widely.