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If the function of f(x) has a domain of [4,6] and a range of [2,3], what will the domain and range of g(x) be? $\frac{1}{2}g\left(x\right)+2=2f\left(x-3\right)$12 g(x)+2=2ƒ (x−3)

3 years ago

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[ (x3)2]= (x3)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 (x3)

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 (x3)→4×ƒ (x3)

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×ƒ (x3)→4×ƒ (x3)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:  $\left[8-4,12-4\right]=\left[4,8\right]$[84,124]=[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] .