To solve the problem of determining which equation represents the function [tex]\( m \)[/tex] after a phase shift to the left by [tex]\(\frac{\pi}{3}\)[/tex] units, we need to understand what a phase shift does to the cosine function.
The general form of a cosine function with a phase shift is given by:
[tex]\[ y = \cos(x - c) \][/tex]
Where [tex]\( c \)[/tex] is the phase shift. A positive [tex]\( c \)[/tex] corresponds to a shift to the right, and a negative [tex]\( c \)[/tex] corresponds to a shift to the left.
In this problem, we are looking for a phase shift to the left by [tex]\(\frac{\pi}{3}\)[/tex]. Therefore, our phase shift [tex]\( c \)[/tex] is negative:
[tex]\[ c = -\frac{\pi}{3} \][/tex]
So, the transformed cosine function should be:
[tex]\[ y = \cos\left(x - (-\frac{\pi}{3})\right) = \cos\left(x + \frac{\pi}{3}\right) \][/tex]
Now, let's compare this to the given options:
A. [tex]\( m(x) = 3 \cos(x - \pi) \)[/tex]
- This function represents a cosine function with a shift to the right by [tex]\(\pi\)[/tex] units and a vertical stretch by a factor of 3. This does not match a shift to the left by [tex]\(\frac{\pi}{3}\)[/tex].
B. [tex]\( m(x) = \cos(3x - \pi) \)[/tex]
- This function represents a cosine function with an argument scaled by 3 and then shifted to the right by [tex]\(\pi\)[/tex] units. It does not match a shift to the left.
C. [tex]\( m(x) = \cos\left(\frac{1}{3}x - \pi\right) \)[/tex]
- This function reduces the frequency by a factor of 3 and shifts to the right by [tex]\(\pi\)[/tex]. This again does not match a shift to the left by [tex]\(\frac{\pi}{3}\)[/tex].
D. [tex]\( m(x) = \cos(3x + \pi) \)[/tex]
- This function involves a horizontal scaling by a factor of 3 and a shift to the left by [tex]\(\pi\)[/tex]. Since it represents a left shift, let's verify if it aligns with the described transformation:
- Scaling and shifting might seem non-intuitive, but this function inherently adjusts the transformation direction but not to the required [tex]\(\frac{\pi}{3}\)[/tex].
Therefore, the correct transformed equation representing the function [tex]\( m \)[/tex] with the desired phase shift is none of the given options. So, instead of A, B, C, or D:
[tex]\[ \boxed{\cos\left(x + \frac{\pi}{3}\right)} \][/tex]
Is the correct transformed function for the left shift by [tex]\(\frac{\pi}{3}\)[/tex]. Reconsider the presented answers for correctness or typographical misalignment.
