Question

Determine the range of
`√((x-5)(x+3))` without graphing it.

Answer 1

Consider the function
`f(x) = (x-5)(x+3)`. It is a parabola with a vertex below the x-axis and x-intercepts of 5 and -3. The vertex x-coordinate is in the middle of the two x-intercepts, so the vertex x-coordinate is (5+-3)/2 = 1. The y-coordinate of the vertex is obtained by substituiting the x-coordinate of the vettex inside, which gets you -16. Its vertex is at (1,-16) and it opens upward; therefore, (x-5)(x+3) gets you a range of
`y \ge -16`.

But when you take square root, there are no negatives. Anything where the range is negative in (x-5)(x+3) is cut out, and we are left with
`y \ge 0`. Therefore, the range of
`√((x-5)(x+3))` is
`y \ge 0`