If the range of the function \(f(x)=x^2+2x+3\), defined on the real numbers, is \([m,\infty)\), what is the value of \(m\)?

Correct answer: 2

Completing the square gives \(f(x)=x^2+2x+3=(x+1)^2+2\). Since \((x+1)^2\geq 0\), the function has minimum value \(2\), attained at \(x=-1\). Hence its range is \([2,\\infty)\), so \(m=2\). Exam tip: For a quadratic written as \(a(x-h)^2+k\) with \(a>0\), the minimum value is \(k\). The constant term \(3\) is not the minimum value.