The range of the function g(x) = x2 + 12 can be determined by analyzing its components. Since this is a quadratic function, it’s important to note the properties of the parabola it represents.
The term x2 is always non-negative, meaning it has a minimum value of 0 when x = 0. Therefore, the smallest value of g(x) occurs when x2 is at its minimum:
g(0) = 0 + 12 = 12.
As x moves away from 0—either positively or negatively—x2 increases, leading g(x) to also increase without bound. Thus, the function will take on all values greater than or equal to 12.
In interval notation, the range of the function can be expressed as:
[12, ∞)
This indicates that g(x) starts at 12 and continues to increase indefinitely.