The domain of y=g(x) is −1≤x≤11 and the range is −13≤y≤16. Give both the domain and range of the following two functions as inequalities.
(a) y=g(12x)
(b) y= 3g(x)
nvm got it lol
(a) For the function y = g(12x), we need to consider the domain and range of the original function g(x) and apply the necessary transformations.
Domain:
Given that the domain of the original function g(x) is -1 ≤ x ≤ 11, we can find the domain of y = g(12x) by considering the new x-values. Since we have multiplied x by 12, the new x-values will be scaled down by a factor of 1/12. Therefore, the domain of y = g(12x) is:
-1/12 ≤ 12x ≤ 11/12
Range:
The range of y = g(12x) depends on the range of the original function g(x). Since there is no information about the range of g(x), we cannot provide a precise range for y = g(12x) without more information.
(b) For the function y = 3g(x), we need to consider the domain and range of the original function g(x) and apply the necessary transformations.
Domain:
The domain of y = 3g(x) will be the same as the domain of the original function g(x), which is -1 ≤ x ≤ 11.
Range:
The range of y = 3g(x) can be obtained by multiplying the range of the original function g(x) by 3. Given that the range of g(x) is -13 ≤ y ≤ 16, the new range of y = 3g(x) will be:
-39 ≤ y ≤ 48
To find the domain and range of the given functions, we need to consider the operations being applied to the variable and the given domain and range of the original function, g(x).
(a) For the function y = g(12x):
To obtain the domain, we need to solve for the values of x that satisfy the given domain of g(x), which is -1 ≤ x ≤ 11. Since we are multiplying x by 12, we divide both sides of the inequality by 12 to find the new domain:
((-1)/12) ≤ (x/12) ≤ (11/12)
Simplifying gives us:
-1/12 ≤ x/12 ≤ 11/12
Therefore, the domain for y = g(12x) is -1/12 ≤ x/12 ≤ 11/12.
Next, to determine the range, we analyze the given range of g(x), which is -13 ≤ y ≤ 16. In this case, since we're only applying a function to the x-values, the range remains unchanged. So the range for y = g(12x) is -13 ≤ y ≤ 16.
Therefore, the domain for the function y = g(12x) is -1/12 ≤ x/12 ≤ 11/12, and the range is -13 ≤ y ≤ 16.
(b) For the function y = 3g(x):
For the domain, we use the original domain of -1 ≤ x ≤ 11, since we are not performing any operations on the x-values. Therefore, the domain for y = 3g(x) remains as -1 ≤ x ≤ 11.
To determine the range, we consider the given range of g(x), which is -13 ≤ y ≤ 16. In this case, we are multiplying the output of g(x) by 3, which means the new range will be multiplied by 3 as well:
3(-13) ≤ 3y ≤ 3(16)
Simplifying gives us:
-39 ≤ 3y ≤ 48
Dividing both sides by 3:
-13 ≤ y ≤ 16
Thus, the domain for y = 3g(x) is -1 ≤ x ≤ 11, and the range is -13 ≤ y ≤ 16.