Answer :
To determine the function [tex]\( g(x) \)[/tex] that results from applying the specified transformations to the initial function [tex]\( f(x) = 9 \cos \left(x - \frac{\pi}{2}\right) + 3 \)[/tex], let's proceed step-by-step.
### Step 1: Translating [tex]\( \frac{\pi}{6} \)[/tex] units left
Translating a function horizontally by [tex]\( \frac{\pi}{6} \)[/tex] units to the left means we need to adjust the argument of the cosine function. The horizontal shift is applied inside the cosine function by adding [tex]\( \frac{\pi}{6} \)[/tex]:
[tex]\[ f(x) = 9 \cos \left(x - \frac{\pi}{2}\right) + 3 \][/tex]
[tex]\[ g(x) = 9 \cos \left((x + \frac{\pi}{6}) - \frac{\pi}{2}\right) + 3 \][/tex]
Simplifying the expression inside the cosine function:
[tex]\[ g(x) = 9 \cos \left(x + \frac{\pi}{6} - \frac{\pi}{2}\right) + 3 \][/tex]
[tex]\[ g(x) = 9 \cos \left(x - \frac{\pi}{2} + \frac{\pi}{6}\right) + 3 \][/tex]
[tex]\[ g(x) = 9 \cos \left(x - \frac{\pi}{3}\right) + 3 \][/tex]
### Step 2: Translating 4 units up
Translating a function vertically by 4 units up means adding 4 to the entire value of the function:
[tex]\[ g(x) = 9 \cos \left(x - \frac{\pi}{3}\right) + 3 + 4 \][/tex]
[tex]\[ g(x) = 9 \cos \left(x - \frac{\pi}{3}\right) + 7 \][/tex]
After applying both transformations, we obtain the final expression for [tex]\( g(x) \)[/tex]:
[tex]\[ g(x) = 9 \cos \left(x - \frac{\pi}{3}\right) + 7 \][/tex]
### Conclusion:
The correct equation that represents [tex]\( g(x) \)[/tex] is:
[tex]\[ g(x) = 9 \cos \left(x - \frac{\pi}{3}\right) + 7 \][/tex]
Thus, the correct option is:
[tex]\[ \boxed{2} \][/tex]
Option 2, [tex]\( g(x) = 9 \cos (x - \frac{\pi}{3}) + 7 \)[/tex], is the equation that represents [tex]\( g(x) \)[/tex] after translating [tex]\( f(x) \)[/tex] both [tex]\( \frac{\pi}{6} \)[/tex] units left and 4 units up.
### Step 1: Translating [tex]\( \frac{\pi}{6} \)[/tex] units left
Translating a function horizontally by [tex]\( \frac{\pi}{6} \)[/tex] units to the left means we need to adjust the argument of the cosine function. The horizontal shift is applied inside the cosine function by adding [tex]\( \frac{\pi}{6} \)[/tex]:
[tex]\[ f(x) = 9 \cos \left(x - \frac{\pi}{2}\right) + 3 \][/tex]
[tex]\[ g(x) = 9 \cos \left((x + \frac{\pi}{6}) - \frac{\pi}{2}\right) + 3 \][/tex]
Simplifying the expression inside the cosine function:
[tex]\[ g(x) = 9 \cos \left(x + \frac{\pi}{6} - \frac{\pi}{2}\right) + 3 \][/tex]
[tex]\[ g(x) = 9 \cos \left(x - \frac{\pi}{2} + \frac{\pi}{6}\right) + 3 \][/tex]
[tex]\[ g(x) = 9 \cos \left(x - \frac{\pi}{3}\right) + 3 \][/tex]
### Step 2: Translating 4 units up
Translating a function vertically by 4 units up means adding 4 to the entire value of the function:
[tex]\[ g(x) = 9 \cos \left(x - \frac{\pi}{3}\right) + 3 + 4 \][/tex]
[tex]\[ g(x) = 9 \cos \left(x - \frac{\pi}{3}\right) + 7 \][/tex]
After applying both transformations, we obtain the final expression for [tex]\( g(x) \)[/tex]:
[tex]\[ g(x) = 9 \cos \left(x - \frac{\pi}{3}\right) + 7 \][/tex]
### Conclusion:
The correct equation that represents [tex]\( g(x) \)[/tex] is:
[tex]\[ g(x) = 9 \cos \left(x - \frac{\pi}{3}\right) + 7 \][/tex]
Thus, the correct option is:
[tex]\[ \boxed{2} \][/tex]
Option 2, [tex]\( g(x) = 9 \cos (x - \frac{\pi}{3}) + 7 \)[/tex], is the equation that represents [tex]\( g(x) \)[/tex] after translating [tex]\( f(x) \)[/tex] both [tex]\( \frac{\pi}{6} \)[/tex] units left and 4 units up.
