Domain and range

The domain is the set of permitted inputs. The range is the set of outputs actually produced from that domain. A stated interval can restrict an otherwise valid expression. Exclude inputs making a denominator zero; a real square root requires a nonnegative radicand.

Example

For \(f(x)=x^2+1\) with \(-2\le x\le3\), the smallest output is 1 at \(x=0\), and the largest is 10 at \(x=3\). The range is \(1\le f(x)\le10\). Checking only the two domain endpoints would miss the minimum.

For \(g(x)=\dfrac2{x-4}\), input 4 is forbidden. Its output cannot be zero: the numerator is nonzero. For \(h(x)=\sqrt{x-2}\), the domain is \(x\ge2\) and range is \(h(x)\ge0\).