https://file.aiquickdraw.com/mathbot/file/1725834575142crtuia7z.jpg

Let's analyze the function provided in the image:

### Function:
\[ f(x) = 2x^2 + 2 \]

### Interval:
The function is defined over the interval \(-2 \leq x \leq 5\).

### Steps to find the range:
1. Since this is a quadratic function \(2x^2 + 2\), the graph will be a parabola opening upwards (because the coefficient of \(x^2\) is positive).
2. To find the range, we need to check the values of \(f(x)\) at the boundaries of the interval and the vertex of the parabola, since the vertex will give us the minimum value (as the parabola opens upwards).

#### Finding the vertex:
The vertex of a parabola in the form \(f(x) = ax^2 + bx + c\) is given by the formula:
\[
x_{vertex} = -\frac{b}{2a}
\]
For \(f(x) = 2x^2 + 2\), \(a = 2\) and \(b = 0\), so:
\[
x_{vertex} = -\frac{0}{2(2)} = 0
\]
Thus, the vertex occurs at \(x = 0\).

#### Evaluating the function at key points:
- At \(x = -2\):
  \[
  f(-2) = 2(-2)^2 + 2 = 2(4) + 2 = 8 + 2 = 10
  \]
- At \(x = 5\):
  \[
  f(5) = 2(5)^2 + 2 = 2(25) + 2 = 50 + 2 = 52
  \]
- At the vertex, \(x = 0\):
  \[
  f(0) = 2(0)^2 + 2 = 2
  \]

#### Conclusion:
- The minimum value of the function occurs at \(x = 0\) and is \(f(0) = 2\).
- The maximum value of the function occurs at \(x = 5\) and is \(f(5) = 52\).

### Range:
Thus, the range of \(f(x)\) over the interval \(-2 \leq x \leq 5\) is:
\[
2 \leq f(x) \leq 52
\]

### Answer:
The correct choice based on the options provided is:
\[
10 \leq f(x) < 52
\]

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Would you like more details or have any questions? 

Here are some related questions to explore:
1. How do you determine the direction of a parabola?
2. What is the effect of increasing the coefficient of \(x^2\) on the range of the function?
3. How can we find the vertex of a parabola if the \(b\)-term is non-zero?
4. What are the general properties of quadratic functions?
5. How does changing the interval affect the range of a quadratic function?

**Tip:** Always evaluate the function at the endpoints and vertex when finding the range of quadratic functions over a closed interval.

What is the range of the function f(x) = 2x^2 + 2 over the interval of -2 ≤ x < 5?

Determine the range of the quadratic function f(x) = 2x^2 + 2 over the interval -2 ≤ x ≤ 5 by evaluating the function at key points, including the vertex. This problem covers quadratic function properties and is suitable for high school students in Grades 10-12.

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