Natural domain of functions

Question

Fill the correct numbers in the blanks

The natural domain of the function f(x)=√35-2x-x² ia the closed interval [_?__ , __?__]

Answer 1

We want the radicand, ie the polynomial function under the sq root, to be positive. To find that interval, we can factor it:

f(x) = √-(x² + 2x - 35) = √[-(x + 7)(x - 5)]. The quadratic (even degree) with a negative leading coefficient will start in QIII (ie with negative y-values), and finish in Q IV (also negative y-values). It changes signs at x = -7 and again at x = 5. So it will be non-negative (graphing above the x-axis) for - 7 ≤ x ≤ 5, which is its domain.