Answer to Question #251158 in Calculus for Shawty

Question #251158

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


1
Expert's answer
2021-10-14T18:05:19-0400

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)\n=a((x+\\frac{b}{2a})^2-\\frac{b^2}{4a^2}+\\frac{c}a)\n=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)."


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