Question

The range of the function \( f(x) = x^2 + 2x + 2 \) is:

• A. \( (1, \infty) \)
• B. \( [2, \infty) \)
• C. \( (0, \infty) \)
• D. \( [1, \infty) \)
• E. \( (-\infty, \infty) \)

Hint:

For a parabola opening upwards ($a > 0$), the range always starts at the y-coordinate of the vertex. You can find the x-coordinate of the vertex quickly using $x = -b/2a$.

Answer 1

The Correct Option is D

Concept: The range of a quadratic function \( f(x) = ax^2 + bx + c \) with \( a > 0 \) is \( [\frac{-\mathcal{D}}{4a}, \infty) \), where \( \mathcal{D} = b^2 - 4ac \). Alternatively, completing the square reveals the vertex and the minimum value of the parabola.

Step 1: Complete the square.
Rewrite the function to identify its vertex: \[ f(x) = x^2 + 2x + 1 + 1 \] \[ f(x) = (x + 1)^2 + 1 \]

Step 2: Identify the minimum value.
Since \( (x+1)^2 \ge 0 \) for all real \( x \), the minimum value occurs when \( (x+1)^2 = 0 \): \[ f(x)_{min} = 0 + 1 = 1 \]

Step 3: Determine the range.
The function starts at 1 and increases indefinitely as \( x \) moves away from \(-1\). Thus, the range is \( [1, \infty) \).