Answered By Leonardo F

This is a question that involves the concept of translations and transformations in a function. If we isolate the formula for g(x), we will have:

  $g\left(x\right)=2\left[2f\left(x-3\right)-2\right]=4f\left(x-3\right)-4$g(x)=2[2ƒ (x−3)−2]=4ƒ (x−3)−4  

Then, we have to check what happens to the original funtion f(x):

1) We performed a horizontal shift 3 units to the right:  $f\left(x\right)\rightarrow f\left(x-3\right)$ƒ (x)→ƒ (x−3) 

2) We performed a vertical strech about the x-axis by a factor of 4:  $f\left(x-3\right)\rightarrow4\times f\left(x-3\right)$ƒ (x−3)→4׃ (x−3) 

3) We performed a vertical translation of 4 units down:  $4\times f\left(x-3\right)\rightarrow4\times f\left(x-3\right)-4$4׃ (x−3)→4׃ (x−3)−4 

Any horizontal shift (to the right or to the left) doesn't change the range, only the domain of the function. Hence, our new domain for g(x) will be:  $\left[4+3,6+3\right]=\left[7,9\right]$[4+3,6+3]=[7,9] 

In the second and third transformations, we don't change the domain, only the range of the function, because the graph changes its shape in the y-axis, not the x-axis. Hence, given the second transformation, our new range will be:  $\left[2\times4,3\times4\right]=\left[8,12\right]$[2×4,3×4]=[8,12] 

Finally, we have to account for the third transformation (4 units down). Hence, our new range will be:  $\left[8-4,12-4\right]=\left[4,8\right]$[8−4,12−4]=[4,8] 

In summary, the domain of the function g(x) will be  $\left[7,9\right]$[7,9] and its range will be  $\left[4,8\right]$[4,8] .