Answer :
Sure, let’s carefully analyze each function to determine which one is not a sinusoid.
### Definition of a Sinusoid
A sinusoid is a mathematical curve that describes a smooth, periodic oscillation. It is a type of continuous wave and can be represented by sine or cosine functions, which have the general forms:
[tex]\[ y = A \sin(Bx + C) + D \][/tex]
[tex]\[ y = A \cos(Bx + C) + D \][/tex]
where [tex]\( A \)[/tex], [tex]\( B \)[/tex], [tex]\( C \)[/tex], and [tex]\( D \)[/tex] are constants.
### Analysis of Each Option
#### Option A: [tex]\( y = \sin x \)[/tex]
- This function is clearly a sine function, which is by definition a sinusoid.
- It oscillates periodically between [tex]\(-1\)[/tex] and [tex]\(1\)[/tex].
#### Option B: [tex]\( y = \sqrt{x} \)[/tex]
- This function represents the square root of [tex]\( x \)[/tex].
- It is not periodic. As [tex]\( x \)[/tex] increases, [tex]\( y = \sqrt{x} \)[/tex] monotonically increases without oscillation.
- Therefore, it does not exhibit the characteristics of a sinusoid.
#### Option C: [tex]\( y = \cos x \)[/tex]
- This function is clearly a cosine function, which is a type of sinusoid.
- It also oscillates periodically between [tex]\(-1\)[/tex] and [tex]\(1\)[/tex].
#### Option D: None of the above are sinusoids.
- This option would imply that neither [tex]\( \sin x \)[/tex] nor [tex]\( \cos x \)[/tex] are sinusoids, which contradicts their definitions.
- Therefore, this option is incorrect.
### Conclusion
Among the given options, the function that does not qualify as a sinusoid is:
[tex]\[ \boxed{B. \, y = \sqrt{x}} \][/tex]
This function does not exhibit oscillatory or periodic behavior, and thus it is not a sinusoid.
### Definition of a Sinusoid
A sinusoid is a mathematical curve that describes a smooth, periodic oscillation. It is a type of continuous wave and can be represented by sine or cosine functions, which have the general forms:
[tex]\[ y = A \sin(Bx + C) + D \][/tex]
[tex]\[ y = A \cos(Bx + C) + D \][/tex]
where [tex]\( A \)[/tex], [tex]\( B \)[/tex], [tex]\( C \)[/tex], and [tex]\( D \)[/tex] are constants.
### Analysis of Each Option
#### Option A: [tex]\( y = \sin x \)[/tex]
- This function is clearly a sine function, which is by definition a sinusoid.
- It oscillates periodically between [tex]\(-1\)[/tex] and [tex]\(1\)[/tex].
#### Option B: [tex]\( y = \sqrt{x} \)[/tex]
- This function represents the square root of [tex]\( x \)[/tex].
- It is not periodic. As [tex]\( x \)[/tex] increases, [tex]\( y = \sqrt{x} \)[/tex] monotonically increases without oscillation.
- Therefore, it does not exhibit the characteristics of a sinusoid.
#### Option C: [tex]\( y = \cos x \)[/tex]
- This function is clearly a cosine function, which is a type of sinusoid.
- It also oscillates periodically between [tex]\(-1\)[/tex] and [tex]\(1\)[/tex].
#### Option D: None of the above are sinusoids.
- This option would imply that neither [tex]\( \sin x \)[/tex] nor [tex]\( \cos x \)[/tex] are sinusoids, which contradicts their definitions.
- Therefore, this option is incorrect.
### Conclusion
Among the given options, the function that does not qualify as a sinusoid is:
[tex]\[ \boxed{B. \, y = \sqrt{x}} \][/tex]
This function does not exhibit oscillatory or periodic behavior, and thus it is not a sinusoid.
