The range of function f(x) = `| x +2|` is :

Options

A. R
B. `[0,oo[`
C. `]-oo , 0]`
D. `]0,oo[`

Answer

To find the range of the function \( f(x) = |x + 2| \), we will analyze the expression step by step.

### Step 1: Understand the Absolute Value Function
The absolute value function \( |x| \) is defined as:
- \( |x| = x \) if \( x \geq 0 \)
- \( |x| = -x \) if \( x < 0 \)

### Step 2: Analyze the Function
In our case, we have \( f(x) = |x + 2| \). This means we need to determine when the expression \( x + 2 \) is positive or negative.

### Step 3: Determine Critical Points
Set the inside of the absolute value to zero to find the critical point:
\[ x + 2 = 0 \implies x = -2 \]

### Step 4: Evaluate the Function in Different Intervals
Now we will evaluate \( f(x) \) in two intervals: \( x < -2 \) and \( x \geq -2 \).

1. **For \( x < -2 \)**:
- Here, \( x + 2 < 0 \), so:
\[ f(x) = |x + 2| = -(x + 2) = -x - 2 \]
As \( x \) approaches \(-\infty\), \( f(x) \) approaches \( +\infty \). At \( x = -2 \), \( f(-2) = 0 \).

2. **For \( x \geq -2 \)**:
- Here, \( x + 2 \geq 0 \), so:
\[ f(x) = |x + 2| = x + 2 \]
As \( x \) increases from \(-2\) to \(+\infty\), \( f(x) \) starts from \(0\) (when \( x = -2 \)) and increases to \(+\infty\).

### Step 5: Combine the Results
From both intervals, we see that:
- As \( x \) approaches \(-\infty\), \( f(x) \) can take any positive value.
- At \( x = -2 \), \( f(x) = 0 \).
- As \( x \) approaches \(+\infty\), \( f(x) \) also approaches \(+\infty\).

### Conclusion
Thus, the range of the function \( f(x) = |x + 2| \) is:
\[ [0, +\infty) \]

### Final Answer
The range of the function \( f(x) = |x + 2| \) is \( [0, \infty) \).

---