The range of `f(x) = 2+2x +x^(2)` is

Options

A. `(-oo, oo)`
B. `(-oo, 1)`
C. `[1,oo)`
D. `(0, oo)`

Answer

To find the range of the function \( f(x) = 2 + 2x + x^2 \), we can follow these steps:

### Step 1: Rewrite the function
We start with the function:
\[ f(x) = 2 + 2x + x^2 \]
We can rearrange it to group the terms:
\[ f(x) = x^2 + 2x + 2 \]

### Step 2: Complete the square
To find the minimum value of the quadratic function, we can complete the square:
\[ f(x) = x^2 + 2x + 1 + 1 = (x + 1)^2 + 1 \]
Here, we added and subtracted 1 to complete the square.

### Step 3: Analyze the completed square
The expression \( (x + 1)^2 \) represents a perfect square, which is always non-negative:
\[ (x + 1)^2 \geq 0 \]
Thus, the minimum value of \( (x + 1)^2 \) occurs when \( x + 1 = 0 \) or \( x = -1 \). At this point:
\[ f(-1) = (0)^2 + 1 = 1 \]

### Step 4: Determine the range
Since \( (x + 1)^2 \) can take any non-negative value, the function \( f(x) \) can take values starting from its minimum value of 1 and going to infinity:
\[ f(x) \geq 1 \]
Thus, the range of \( f(x) \) is:
\[ [1, \infty) \]

### Final Answer
The range of the function \( f(x) = 2 + 2x + x^2 \) is:
\[ [1, \infty) \]