Question
What are the domain and range of the real-valued function f(x) = -3 + sqrt 4x-12?
A)The domain is x>3, and the range is f(x)<-3.
B)The domain is x<3, and the range is f(x)<-3.
C)The domain is x>3, and the range is f(x)>-3.
D)The domain is x>3, and the range is all real numbers.
A)The domain is x>3, and the range is f(x)<-3.
B)The domain is x<3, and the range is f(x)<-3.
C)The domain is x>3, and the range is f(x)>-3.
D)The domain is x>3, and the range is all real numbers.
Answers
GPT 3.5
The domain of the function is determined by the restrictions on the square root within the function. Since the square root of a real number must be non-negative, the expression under the square root, 4x - 12, must be greater than or equal to 0:
4x - 12 ≥ 0
4x ≥ 12
x ≥ 3
Therefore, the domain of the function is x ≥ 3.
The range of the function can be found by analyzing the behavior of the function as x approaches infinity and negative infinity. As x approaches negative infinity, the sqrt(4x - 12) approaches sqrt(-∞), which is undefined. Therefore, the function approaches -3. As x approaches positive infinity, the sqrt(4x - 12) approaches infinity, making the function approach infinity as well. The range of the function is all real numbers greater than -3.
Therefore, the correct answer is D) The domain is x ≥ 3, and the range is all real numbers.
4x - 12 ≥ 0
4x ≥ 12
x ≥ 3
Therefore, the domain of the function is x ≥ 3.
The range of the function can be found by analyzing the behavior of the function as x approaches infinity and negative infinity. As x approaches negative infinity, the sqrt(4x - 12) approaches sqrt(-∞), which is undefined. Therefore, the function approaches -3. As x approaches positive infinity, the sqrt(4x - 12) approaches infinity, making the function approach infinity as well. The range of the function is all real numbers greater than -3.
Therefore, the correct answer is D) The domain is x ≥ 3, and the range is all real numbers.