determine the domain of g(f(x)) = sqrt of 12-4x

Answer 1

To determine the domain of g(f(x)) = sqrt(12 - 4x), we first need to consider the domain of the inner function f(x) = 12 - 4x.

The inner function f(x) can take any real number as input, as there are no restrictions on x for the expression 12 - 4x. Therefore, the domain of f(x) is all real numbers.

Next, we need to consider the domain of the outer function g(x) = sqrt(x). The square root function is defined for all non-negative real numbers, i.e., x ≥ 0. If the expression inside the square root is negative, then the function is undefined.

We need to ensure that the expression inside the square root in g(f(x)) = sqrt(12 - 4x) is non-negative. Therefore, we set 12 - 4x ≥ 0 and solve for x:

12 - 4x ≥ 0
4x ≤ 12
x ≤ 3

So, the domain of g(f(x)) = sqrt(12 - 4x) is x ≤ 3.