Find the domain and range of the function.
$f ( x ) = 4 x ^ { 2 } - 64 x + 266$
A) $\text { Domain: } ( - \infty , \infty ) \text {; Range: } [ 0 , \infty )$
B) $\text { Domain: } ( - \infty , \infty ) \text {; Range: } ( - \infty , 8 ]$
C) $\text { Domain: } ( - \infty , \infty ) \text {; Range: } [ 10 , \infty )$
D) $\text { Domain: } ( - \infty , \infty ) \text {; Range: } ( - \infty , 0 ]$

Question

Quizplus · Accepted Answer

The domain of any quadratic function is $(-\infty, \infty)$ because you can input any real number. To find the range, complete the square to find the vertex. The given function is a parabola opening upwards (since the coefficient of $x^2$ is positive), so the minimum value (vertex y-coordinate) will give the start of the range. Completing the square: $f(x) = 4(x^2 - 16x) + 266 = 4[(x - 8)^2 - 64] + 266 = 4(x - 8)^2 - 256 + 266 = 4(x - 8)^2 + 10$. The vertex is at $(8, 10)$, so the minimum y-value is 10, making the range $[10, \infty)$.

Find the Domain and Range of the Function f(x)=4x2−64x+266f ( x ) = 4 x ^ { 2 } - 64 x + 266f(x)=4x2−64x+266

Find the Domain and Range of the Function $f ( x ) = 4 x ^ { 2 } - 64 x + 266$