The function f(x) = x² – 2 is a quadratic function, which is a type of polynomial function that can be represented graphically as a parabola. To determine its range, we should first identify its vertex and behavior as x approaches positive and negative infinity.
1. **Finding the Vertex**: The standard form of a quadratic function is f(x) = ax² + bx + c. In this case, we have:
- a = 1
- b = 0
- c = -2
The vertex of a parabola given by the formula f(x) = ax² + bx + c can be found using the vertex formula:
x = -b / (2a)
Substituting in our values:
x = -0 / (2 * 1) = 0
Now that we have the x-coordinate of the vertex, we can find the corresponding y-coordinate:
f(0) = (0)² – 2 = -2
Thus, the vertex of the function is at the point (0, -2).
2. **Analyzing Direction**: Since the coefficient of x² (which is a = 1) is positive, the parabola opens upwards. This characteristic indicates that the vertex represents the minimum point of the function.
3. **Determining the Range**: As the parabola opens upwards from the vertex at (0, -2), the output values of the function will start from the vertex coordinate, -2, and will extend to infinity. Therefore, the range of the function can be expressed as:
Range: [-2, ∞)
In conclusion, the function f(x) = x² – 2 has a value of its range starting from -2 and extending infinitely upwards, which we represent mathematically as [-2, ∞).