The domain of a function refers to all possible values of the input or all numbers that the variable x can take. In the function f(x)=2(x+4)²-5, you can see there isn't any restriction on x. There is no denominator where we would worry about division by zero, nor is there a square root where we could only allow non-negative values. Therefore, the domain of this function is all real numbers.
To find the range or the set of possible output values of the function, we consider the behaviour of the function. This function is a shifted and scaled version of a parabola (y = x²), a shape which has a minimum or maximum point: its vertex.
The (x+4)² part is always positive or equal to zero because it is a square. By multiplying this by 2, it will remain positive or zero. Then, when we subtract 5, this will either decrease the value by 5 or, at the minimum, become -5 (when (x+4)²=0).
As x gets large (positive or negative), our function goes to +∞ because we're square it, multiplying by 2, and subtracting 5.
Therefore, the range is -5 to ∞. This signifies all real numbers greater than or equal to -5.