1. **State the problem:** Find the range of the function $F(x) = 2x^2 + 1$. 2. **Recall the formula and properties:** The function is a quadratic function in the form $ax^2 + bx + c$ with $a=2 > 0$, so it opens upwards. 3. **Determine the vertex:** The vertex of $F(x)$ is at $x=0$ because there is no $x$ term (i.e., $b=0$). 4. **Calculate the vertex value:** $$F(0) = 2(0)^2 + 1 = 1$$ 5. **Analyze the range:** Since the parabola opens upwards and the minimum value is at the vertex, the minimum value of $F(x)$ is 1. 6. **Write the range:** The function values are all real numbers greater than or equal to 1. **Final answer:** $$\text{Range} = [1, \infty)$$