Final answer:
The domain of the function f(x) = (x – 3)² + 6 is all real numbers, and the range is all real numbers greater than 6, making the correct answer Option A.
Step-by-step explanation:
The domain and range of a function describe the set of possible input values (domain) and output values (range), respectively. For the function f(x) = (x – 3)² + 6, we can determine the domain and range as follows:
The domain of this function is all real numbers because you can plug any real number into the function for x and it will work properly. There are no restrictions like division by zero or taking the square root of a negative number in this case. Therefore, the correct domain is x is all real numbers.
To find the range, we look at the output values of f(x). The expression (x – 3)² is the square of a real number, which is always non-negative. Adding 6 to this non-negative result means the smallest value f(x) can take is when (x – 3)² is zero, so the minimum value of f(x) is 6. Thus, the function's output can be 6 or any number greater than 6, but never less, making the range y ≥ 6. However, we must strictly indicate that y is greater than 6 and not equal to 6 because the function's square term can never be negative and will always add some positive value to 6 for any x ≠ 3. Consequently, the correct range is y > 6.
The correct answer for both domain and range is Option A: Domain: x is all real numbers, Range: y .