The range of the functions f(x) = x² + 2x + 2 is

• (1, ∞)
• (2, ∞)
• (0, ∞)
• [1, ∞)

To find the range of the function f(x) = x² + 2x + 2, we can start by rewriting it in a more useful form. This function is a quadratic equation, and we can complete the square.

Completing the Square

We can express f(x) as follows:

f(x) = (x² + 2x + 1) + 1 = (x + 1)² + 1

Analyzing the Function

The term (x + 1)² is always non-negative (it’s zero or positive) because squares of real numbers cannot be negative. The minimum value occurs when (x + 1)² = 0, which happens at x = -1.

Finding the Minimum Value

At x = -1:

f(-1) = 0 + 1 = 1

Determining the Range

Since (x + 1)² can take any value from 0 to ∞, adding 1 shifts the entire range up by 1. Therefore, the smallest value of f(x) is 1, and it can increase indefinitely.

Final Range

The range of the function f(x) = x² + 2x + 2 is:

• [1, ∞)