Question

Can a quadratic function have a range of (-infinity, infinity) Justify your answer

Answer 1

The quadratic function is of the form $f(x)=ax^2+bx+c,$ where $a\ne 0.$ It follows that $f(x)=a(x^2+\frac{b}ax+\frac{c}a) =a((x+\frac{b}{2a})^2-\frac{b^2}{4a^2}+\frac{c}a) =a(x+\frac{b}{2a})^2+c-\frac{b^2}{4a}.$

If $a>0,$ then the range of $f$ is $[c-\frac{b^2}{4a},+\infty).$ If $a<0,$ then the range of $f$ is $(-\infty,c-\frac{b^2}{4a}].$

We conclude that a quadratic function can not have a range of $(-\infty, +\infty).$