The function g(x) = x² + 12 is a quadratic function, where the term x² determines its shape. Since the coefficient of x² is positive, the parabola opens upwards.
To find the range of this function, we first need to determine its vertex. The vertex of a parabola defined by the function y = ax² + bx + c is located at the x-coordinate:
x = -b/(2a>,
In our case, since there is no linear term (bx), b is 0, and a is 1. Plugging these values into the vertex formula:
x = -0/(2*1) = 0
Now, substituting x = 0 back into the function to find the corresponding y-value (the minimum of the function):
g(0) = 0² + 12 = 12
This result shows that the minimum value of g(x) is 12, which occurs at x = 0. Since the parabola opens upwards and there are no restrictions on x, the function can take any value greater than or equal to 12.
Thus, we conclude that the range of the function g(x) = x² + 12 is:
Range: [12, ∞)