What is the range of the function \(y = x^2\)?

To determine the range of the function (y = x^2), it is essential to understand the behavior of the function as (x) varies over all real numbers. The function (y = x^2) is a quadratic function that opens upwards, forming a parabola.

As (x) takes on any real number value, (y) calculates the square of that value. The output of squaring any real number is always non-negative because the square of a positive number is positive and the square of zero is zero. Moreover, for any negative value of (x), squaring it will also yield a non-negative result, as the product of two negative numbers is positive.

Thus, the minimum value of (y) occurs when (x = 0), yielding (y = 0). As (x) moves away from zero in either direction (positive or negative), (y) increases without bound. Therefore, the possible outputs (or range) of the function start from 0 and extend infinitely upwards.

This behavior leads to the conclusion that the range of (y = x^2) is all values from 0 to infinity, inclusive of 0. This is