To determine which of the given options is a sinusoid, it's important to understand the characteristics of a sinusoidal function. A sinusoidal function is a function that describes a smooth periodic oscillation, which typically can be represented in the form of either [tex]\( y = A \sin(Bx + C) \)[/tex] or [tex]\( y = A \cos(Bx + C) \)[/tex]. These functions produce wave-like graphs called sine and cosine waves.
Let's analyze each option individually to see if it matches the form of a sinusoidal function:
Option A: [tex]\( y = \sqrt[3]{x} \)[/tex]
- This function represents the cube root of [tex]\( x \)[/tex].
- The graph of [tex]\( y = \sqrt[3]{x} \)[/tex] is not periodic and does not resemble a sine or cosine wave. It's a curve that grows slowly in both the positive and negative directions but does not oscillate.
Option B: [tex]\( y = \sin x \)[/tex]
- This function is directly in the form of a sine wave.
- The graph of [tex]\( y = \sin x \)[/tex] is a smooth periodic oscillation, repeating every [tex]\( 2\pi \)[/tex].
- This matches the definition of a sinusoidal function perfectly.
Option C: [tex]\( x^2 + y^2 = 1 \)[/tex]
- This equation represents a circle with a radius of 1, centered at the origin.
- The graph is not a waveform; instead, it forms a circular shape.
- Therefore, it does not meet the criteria of being a sinusoidal function.
Option D: [tex]\( y = [x] \)[/tex]
- This represents the greatest integer function, also known as the floor function.
- The graph of [tex]\( y = [x] \)[/tex] is a step function, increasing by 1 at each integer value of [tex]\( x \)[/tex].
- This is neither smooth nor periodic in the sense of a sine or cosine wave and does not match the characteristics of a sinusoidal function.
Based on this analysis, the only function that is sinusoidal is:
Option B: [tex]\( y = \sin x \)[/tex]
Therefore, the correct choice is B.
