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Kipish [7]
4 years ago
10

Determine the range of

x+3)}" alt="\sqrt{(x-5)(x+3)}" align="absmiddle" class="latex-formula"> without graphing it.
Mathematics
1 answer:
mixas84 [53]4 years ago
4 0

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 \sqrt{(x-5)(x+3)} is y \ge 0

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