If the domain of y=g(x) is D = [-2, 6] and the range is R = [-8,10], find the domain and range of the function y=|g(2x) | +1
D = [-1,3], R = [-7,11]
To find the domain and range of the function y = |g(2x)| + 1, we need to consider two things: the domain and range of the function g(x) and the effects of the absolute value and the addition of 1.
1. Domain of g(x):
Given that the domain of g(x) is D = [-2, 6], we need to consider how this affects the domain of the function y = |g(2x)| + 1.
Since we have g(2x) inside the absolute value, it means that the input to g(x) is multiplied by 2 before being passed into the function g. To find the domain of the function y = |g(2x)| + 1, we need to determine the range of values that 2x can take within the domain [-2, 6].
To do this, we divide the domain of g(x) by 2, as it is inside the argument. So, we have -2/2 = -1 and 6/2 = 3. Therefore, the new domain for 2x is [-1, 3].
2. Effects of the absolute value and addition of 1:
After determining the new domain for 2x, we need to consider the effects of the absolute value and the addition of 1.
The absolute value function |z| takes any input z and returns its non-negative value. So, if g(2x) results in a negative value, the absolute value will convert it to a positive value. Therefore, we don't need to worry about negative values affecting the domain.
The addition of 1 shifts the range of the function by 1 unit upwards. Thus, the range for the function y = |g(2x)| + 1 will be the range of |g(2x)| shifted upward by 1 unit.
3. Determining the range of |g(2x)|:
To find the range of |g(2x)|, we need to consider the range of g(2x) and apply the absolute value function.
Given that the range of g(x) is R = [-8, 10], we divide the range by 2, as 2x is inside the argument. So, we have -8/2 = -4 and 10/2 = 5. Therefore, the new range for g(2x) is [-4, 5].
Applying the absolute value function to this new range, we have [0, 5], as the absolute value of any number is always non-negative.
4. Shifting the range upward by 1 unit:
Finally, we shift the range [0, 5] upward by 1 unit to account for the addition of 1 in the function y = |g(2x)| + 1. This results in the new range being [1, 6].
Therefore, the domain of the function y = |g(2x)| + 1 is [-1, 3], and the range is [1, 6].