Final answer:
The correct option for the domain and range of the given function f(x) = (x - 3) with an absolute value sign and plus 6 is Option 1, which states that the domain is all real numbers and the range is all numbers greater than or equal to 6.
Step-by-step explanation:
The function provided f(x) = (x - 3) with an absolute value sign around (x - 3) and plus 6 at the end is a transformation of the basic absolute value function. The absolute value function by itself, which is |x|, can take any real number as an input, meaning its domain is all real numbers. In this case, the subtraction of 3 inside the absolute value does not change the domain, because we can still input any real number and get a result. The addition of 6 on the outside shifts the graph upwards, but it also does not affect the domain.
For the range, since absolute values are always non-negative and we are adding 6 to it, the smallest value f(x) can take is when x is 3. At x = 3, (x-3) becomes zero, so f(x) becomes |0| + 6, which is 6. Therefore, the function can never be less than 6, making the range y ≥ 6.
With this in mind, Option 1 is correct: Domain: x , Range: y .