Find the range of `f(x)=x^2 +2`

Answer

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

### Step 1: Rewrite the function in terms of \( y \)
We start by setting \( f(x) = y \):
\[ y = x^2 + 2 \]

### Step 2: Rearrange the equation
Next, we rearrange the equation to isolate \( x^2 \):
\[ x^2 = y - 2 \]

### Step 3: Determine the conditions for \( x^2 \)
Since \( x^2 \) is always non-negative (i.e., \( x^2 \geq 0 \)), we can set up the following inequality:
\[ y - 2 \geq 0 \]
This simplifies to:
\[ y \geq 2 \]

### Step 4: Identify the range
From the inequality \( y \geq 2 \), we can conclude that the smallest value of \( y \) is 2, and it can take any value greater than 2. Therefore, the range of the function is:
\[ \text{Range} = [2, \infty) \]

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

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