given the domain of g(x) is (-3,5) and the range of g(x) (-10, 10) find the domain and range of the transformed function 4g(x-7)+12
To find the domain and range of the transformed function 4g(x-7)+12, we need to consider two things: the transformations applied to the original function g(x), and the restrictions on the input values of x based on the original domain.
Let's analyze the transformations step by step:
1. Horizontal Translation (Shift):
The original function g(x) is shifted 7 units to the right, resulting in g(x-7).
This means that all x-values in the original domain (-3, 5) are shifted by 7 units to the right. Therefore, the new domain will be (-3 + 7, 5 + 7) = (4, 12).
2. Vertical Stretch/Compression and Vertical Translation:
The original function g(x) is multiplied by 4, which stretches the vertical dimension of the graph. Then, it is shifted upward by 12 units.
These transformations do not affect the domain of the function, but they do impact the range.
To determine the new range, we consider the original range (-10, 10) and apply the vertical transformations. Since the original function is stretched by a factor of 4, the range is also stretched vertically. So, the new range will be (-10 * 4 + 12, 10 * 4 + 12) = (-28, 52).
Therefore, the domain of the transformed function 4g(x-7)+12 is (4, 12), and the range is (-28, 52).