Question

Find the range of $ f(x) = \sqrt{x^2 + 4x + 4} $.

• A. \( (-\infty, \infty) \)
• B. \( [0, \infty) \)
• C. \( (-2, 2) \)
• D. \( [2, \infty) \)

Hint:

The range of the absolute value function \( |x + 2| \) is always non-negative.

Answer 1

The Correct Option is B

The given function is: \[ f(x) = \sqrt{x^2 + 4x + 4} \] We can simplify the expression inside the square root: \[ x^2 + 4x + 4 = (x + 2)^2 \]
Thus, the function becomes: \[ f(x) = \sqrt{(x + 2)^2} \] Since the square root of a square is the absolute value, we have: \[ f(x) = |x + 2| \] The absolute value function \( |x + 2| \) always yields non-negative values. Therefore, the range of \( f(x) \) is: \[ [0, \infty) \]
Thus, the correct answer is \( [0, \infty) \).