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The image shows a problem asking for the range of the function \( y = \sqrt{x^2 + 4x + 2} - 1 \). Here's how to find the range of this function:

### Step-by-Step Solution:

1. **Analyze the Function Inside the Square Root:**
   The function inside the square root is \( x^2 + 4x + 2 \), which is a quadratic expression. The expression inside the square root must be non-negative, so we start by analyzing this quadratic.

2. **Find the Vertex of the Quadratic:**
   The expression \( x^2 + 4x + 2 \) can be rewritten by completing the square:
   \[
   x^2 + 4x + 2 = (x^2 + 4x + 4) - 4 + 2 = (x + 2)^2 - 2
   \]
   The vertex of this parabola occurs at \( x = -2 \), and the minimum value of the quadratic function is \( (x + 2)^2 - 2 \). At \( x = -2 \), the minimum value is:
   \[
   (x + 2)^2 - 2 = (0)^2 - 2 = -2
   \]

3. **Determine the Minimum Value Under the Square Root:**
   Since the square root function requires non-negative inputs, the expression \( x^2 + 4x + 2 \) must be greater than or equal to zero. But, the lowest value of \( x^2 + 4x + 2 \) is -2, so:
   \[
   \sqrt{x^2 + 4x + 2} \text{ is defined for } x^2 + 4x + 2 \geq 0
   \]

   To ensure this, add 2 to both sides:
   \[
   x^2 + 4x + 2 + 2 \geq 2 \quad \Rightarrow \quad \sqrt{x^2 + 4x + 2} \geq \sqrt{2}
   \]

4. **Range Calculation:**
   Since \( \sqrt{x^2 + 4x + 2} \geq \sqrt{2} \), then the function \( y = \sqrt{x^2 + 4x + 2} - 1 \) will always be at least:
   \[
   y \geq \sqrt{2} - 1
   \]

   As \( x^2 + 4x + 2 \) grows, so does the square root, meaning there is no upper bound on \( y \).

### Conclusion:
The range of the function \( y = \sqrt{x^2 + 4x + 2} - 1 \) is \( [\

The range of the function y = √(x^2 + 4x + 2) - 1 is…

Discover how to find the range of the function y = √(x^2 + 4x + 2) - 1 with a comprehensive step-by-step solution. This problem covers key algebraic concepts, including quadratic equations, completing the square, and understanding the properties of square roots. Ideal for students in Grades 10-12.

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