Question

The function f(x)\:=\:x^{\frac{1}{2}} is transformed to get function w. w(x)\:=\:\textrm{-}(3x)^{\frac{1}{2}}\:-\:4 what are the domain and the range of function w?

a) Domain: All real numbers, Range: All real numbers
b) Domain: x ≥ 0 , Range: y ≤-4
c) Domain: x ≤ 0 , Range: y ≤-4
d) Domain: x ≤ 0 , Range: All real numbers

Answer 1

The domain of the transformed function w(x) = - (3x)½ - 4 is x ≥ 0, and the range is y ≤ -4, as the function involves a reflection over the x-axis, a horizontal scaling, and a downward translation.

Step-by-step explanation:

The transformation of the function f(x) = x½ to w(x) = - (3x)½ - 4 involves a reflection, scaling, and translation. First, the negative sign reflects the square root function over the x-axis. Then, the coefficient 3 inside the square root scales the function horizontally by a factor of ⅓. Finally, subtracting 4 shifts the function down by 4 units.

To determine the domain and range, we observe that since x is within a square root, x must be non-negative to ensure the result is real. Therefore, the domain is all real numbers x ≥ 0. The transformation then reflects the square root graph vertically and shifts it downwards, so w(x) will always be less or equal to -4. This makes the range y ≤ -4. Hence the correct domain and range are: Domain: x ≥ 0, Range: y ≤ -4.