The function f(x) = x² – 2 is a quadratic function, which means its graph is a parabola. The general form of a quadratic function is f(x) = ax² + bx + c, where ‘a’, ‘b’, and ‘c’ are constants.
In this case, ‘a’ is equal to 1 (which is positive), indicating that the parabola opens upwards. The vertex of the parabola can be found using the formula for the vertex of a quadratic function, which is given by the point (h, k) where:
- h = -b/(2a)
- k = f(h)
Here, since there is no ‘b’ term in our function, we can see that b = 0. Therefore, h = 0/(2*1) = 0. Now to find k:
k = f(0) = (0)² – 2 = -2.
This means the vertex of the parabola is at the point (0, -2). Because the parabola opens upwards, the minimum value of the function occurs at the vertex, which is -2.
As x moves away from 0 in either direction (positive or negative), the value of f(x) will continue to increase without bound. Therefore, the function will take values starting from -2 and going up to positive infinity.
In conclusion, the range of the function f(x) = x² – 2 is:
- Range: [-2, ∞)