Source: Extra Practice
Find the domain and range of the real-valued function f(x) = |x - 4| + 3.
Options
Option A is correct
Domain: R, Range: [3, ∞)
Option B
Domain: [4, ∞), Range: [3, ∞)
Option C
Domain: R, Range: R
Option D
Domain: R, Range: [0, ∞)
Explanation
Correct Option: The modulus function |x - 4| is defined for all real numbers, so the Domain is R. Since the minimum value of any absolute value expression |a| is 0, the minimum value of |x - 4| is 0 (at x=4). Adding 3 to this minimum value gives 0 + 3 = 3. Therefore, the function values range from 3 to infinity, making the Range [3, ∞). Incorrect Options: Option 2 incorrectly restricts the domain; Option 3 is wrong because the modulus function cannot output negative values in this context; Option 4 fails to account for the vertical shift of +3.