Final answer:
For the cube root function g(x) = ∛[3]{x+8} - 4, the root occurs at x=56, the y-intercept is at (0, -2), and there is no specified inflection point as the function's rate of change of slope is constant.
Step-by-step explanation:
Finding the Root, Y-Intercept, and Inflection Point of a Cube Root Function
To find the characteristics of the cube root function g(x) = ∛[3]{x+8} - 4, we need to examine its formula. The root of the function is the value of x for which g(x) = 0. Setting the function equal to zero and solving:
0 = ∛[3]{x+8} - 4
4 = ∛[3]{x+8}
64 = x + 8
x = 56
The y-intercept is found by setting x to 0 and evaluating the function:
g(0) = ∛[3]{0+8} - 4
g(0) = ∛[3]{8} - 4
g(0) = 2 - 4
g(0) = -2
Thus, the y-intercept is (0, -2).
The inflection point typically occurs where the second derivative changes sign; however, for the cube root function, an inflection point doesn't exist because the rate of change of the slope is constant.
Therefore, the correct answers are: Root: 56, Y-intercept: (0, -2), and no specified Inflection Point.