The function f(x) = x² + 3 is a quadratic function, where the term x² indicates that it is a parabola. Let’s explore how to find its range step by step.
1. **Understand the Parabola’s Shape**: The graph of the function is a parabola that opens upwards because the coefficient of x² is positive.
2. **Find the Vertex**: The vertex of the parabola provides crucial information about its minimum or maximum value. For a function in the form of f(x) = ax² + bx + c, the x-coordinate of the vertex can be calculated using the formula x = -b/(2a). In our case, a = 1 and b = 0, so:
x = -0 / (2 * 1) = 0This tells us that the vertex (the minimum point of the parabola since it opens upwards) is at x = 0.
3. **Calculate the Minimum Value**: Now, we substitute the value of x back into the function to find the corresponding y-value:
f(0) = (0)² + 3 = 3The minimum value of the function occurs at the vertex, where y = 3.
4. **Determine the Range**: Since the parabola opens upwards, the function will reach its minimum value at y = 3 and increase without bound. Thus, the range of the function is:
Range: [3, ∞)
In conclusion, the function f(x) = x² + 3 has a range starting from 3 and goes to infinity, which can be expressed as [3, ∞).