Answer:
D. (z) = (x − 3)¹ – 2
Explanation:
To determine which transformation of the parent square root function will result in the given domain and range, we need to consider the effects of the transformations on the function.
The parent square root function is given by f(x) = √x.
Let's analyze each option and see if it satisfies the given conditions:
A. j(x) = (x + 2)³ + 3
This transformation involves shifting the graph 2 units to the left and 3 units up. However, this does not change the domain of the function, so it does not satisfy the given domain condition.
B. k(x) = (z + 3) – 2
This transformation involves shifting the graph 3 units to the left and 2 units down. Again, this does not change the domain of the function, so it does not satisfy the given domain condition.
C. g(x) = (x − 2)³ + 3
This transformation involves shifting the graph 2 units to the right and 3 units up. However, this does not change the range of the function, so it does not satisfy the given range condition.
D. z(x) = (x − 3)¹ – 2
This transformation involves shifting the graph 3 units to the right and 2 units down. This shift does not affect the domain of the function, but it affects the range. The function z(x) = (x − 3)¹ – 2 starts at y = -2 when x = 3, and it increases as x goes to infinity. Therefore, it satisfies both the given domain and range conditions.
Based on the analysis, the correct transformation that satisfies the given domain and range is option D:
z(x) = (x − 3)¹ – 2
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