What is the range of the function #y = x^2#?

Answer 1

The range is #y>=0#.

Normally, you would complete the square and check the leading coefficient, #a#, to determine the concavity for the comparison sign. However, this function is already in vertex or standard form:

#y=(x-0)^2+0#

So the vertex is #(0,0)# and the leading coefficient is positive; this means the parabola is concave up and the vertex has the minimum value. The minimum value is the bottom of the range of the function.

Semantics is important, the vertex is not the minimum value because it is a point; it merely contains the minimum value.

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Answer 2

Benjamin Achtenberg

The range of the function (y = x^2) is all real numbers greater than or equal to zero. In interval notation, it can be expressed as ([0, +\infty)). This is because any real number squared is non-negative, and all non-negative real numbers can be achieved as the square of some real number. Therefore, the range of (y = x^2) includes all non-negative real numbers.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.