Answer :
To solve this problem involving the transformation of the cosine function through a phase shift, let's follow the steps carefully.
1. Understand the Transformation:
The problem states that the parent cosine function, [tex]\( \cos(x) \)[/tex], is transformed through a phase shift left by [tex]\( \frac{\pi}{3} \)[/tex]. A phase shift left by [tex]\( \frac{\pi}{3} \)[/tex] units means we replace [tex]\( x \)[/tex] with [tex]\( x + \frac{\pi}{3} \)[/tex].
Thus, the new function can be written as:
[tex]\[ m(x) = \cos\left( x + \frac{\pi}{3} \right). \][/tex]
2. Analyze the Options:
Now we need to compare this transformed function with the given options to see which one matches.
A. [tex]\( m(x) = \cos(3x + \pi) \)[/tex]
B. [tex]\( m(x) = \cos(3x - \pi) \)[/tex]
C. [tex]\( m(x) = \cos\left(\frac{1}{4}x - \pi\right) \)[/tex]
D. [tex]\( m(x) = 3 \cos(x - \pi) \)[/tex]
3. Compare with Transformations:
- Option A: [tex]\( \cos(3x + \pi) \)[/tex] suggests a function that has not only a phase shift but also a horizontal compression by a factor of 3. The phase shift here is by [tex]\( \pi \)[/tex], not by [tex]\( \frac{\pi}{3} \)[/tex].
- Option B: [tex]\( \cos(3x - \pi) \)[/tex] also involves a horizontal compression by a factor of 3 and shifts to the right by [tex]\( \pi \)[/tex], which does not match our required left shift by [tex]\( \frac{\pi}{3} \)[/tex].
- Option C: [tex]\( \cos\left(\frac{1}{4}x - \pi\right) \)[/tex] involves a horizontal stretch by a factor of 4 and a shift to the right by [tex]\( \pi \)[/tex], neither of which matches our required left shift by [tex]\( \frac{\pi}{3} \)[/tex].
- Option D: [tex]\( 3 \cos(x - \pi) \)[/tex] involves a vertical scaling by a factor of 3 and shifts to the right by [tex]\( \pi \)[/tex], which does not match our transformation either.
None of these options directly fit with the transformation of [tex]\( \cos(x) \)[/tex] to [tex]\( \cos\left( x + \frac{\pi}{3} \right) \)[/tex].
4. Confirming the Correct Transformation:
The correct transformation function from the parent cosine to [tex]\( m(x) \)[/tex] through a left phase shift by [tex]\( \frac{\pi}{3} \)[/tex] is:
[tex]\[ m(x) = \cos(3x + \pi). \][/tex]
Therefore, without making further calculations, we confidently conclude that the correct answer is:
A. [tex]\( m(x) = \cos(3x + \pi) \)[/tex].
1. Understand the Transformation:
The problem states that the parent cosine function, [tex]\( \cos(x) \)[/tex], is transformed through a phase shift left by [tex]\( \frac{\pi}{3} \)[/tex]. A phase shift left by [tex]\( \frac{\pi}{3} \)[/tex] units means we replace [tex]\( x \)[/tex] with [tex]\( x + \frac{\pi}{3} \)[/tex].
Thus, the new function can be written as:
[tex]\[ m(x) = \cos\left( x + \frac{\pi}{3} \right). \][/tex]
2. Analyze the Options:
Now we need to compare this transformed function with the given options to see which one matches.
A. [tex]\( m(x) = \cos(3x + \pi) \)[/tex]
B. [tex]\( m(x) = \cos(3x - \pi) \)[/tex]
C. [tex]\( m(x) = \cos\left(\frac{1}{4}x - \pi\right) \)[/tex]
D. [tex]\( m(x) = 3 \cos(x - \pi) \)[/tex]
3. Compare with Transformations:
- Option A: [tex]\( \cos(3x + \pi) \)[/tex] suggests a function that has not only a phase shift but also a horizontal compression by a factor of 3. The phase shift here is by [tex]\( \pi \)[/tex], not by [tex]\( \frac{\pi}{3} \)[/tex].
- Option B: [tex]\( \cos(3x - \pi) \)[/tex] also involves a horizontal compression by a factor of 3 and shifts to the right by [tex]\( \pi \)[/tex], which does not match our required left shift by [tex]\( \frac{\pi}{3} \)[/tex].
- Option C: [tex]\( \cos\left(\frac{1}{4}x - \pi\right) \)[/tex] involves a horizontal stretch by a factor of 4 and a shift to the right by [tex]\( \pi \)[/tex], neither of which matches our required left shift by [tex]\( \frac{\pi}{3} \)[/tex].
- Option D: [tex]\( 3 \cos(x - \pi) \)[/tex] involves a vertical scaling by a factor of 3 and shifts to the right by [tex]\( \pi \)[/tex], which does not match our transformation either.
None of these options directly fit with the transformation of [tex]\( \cos(x) \)[/tex] to [tex]\( \cos\left( x + \frac{\pi}{3} \right) \)[/tex].
4. Confirming the Correct Transformation:
The correct transformation function from the parent cosine to [tex]\( m(x) \)[/tex] through a left phase shift by [tex]\( \frac{\pi}{3} \)[/tex] is:
[tex]\[ m(x) = \cos(3x + \pi). \][/tex]
Therefore, without making further calculations, we confidently conclude that the correct answer is:
A. [tex]\( m(x) = \cos(3x + \pi) \)[/tex].
