A. To show "half" of the graph, we can restrict the domain to only include values of x that result in non-negative values of y. This can be done by setting the expression inside the square brackets to be greater than or equal to zero and solving for x:
3 * (x + 2) ^ 2 - 7 ≥ 0
(x + 2) ^ 2 ≥ 7/3
|x + 2| ≥ sqrt(7/3)
x + 2 ≤ -sqrt(7/3) or x + 2 ≥ sqrt(7/3)
x ≤ -2 - sqrt(7/3) or x ≥ -2 + sqrt(7/3)
So the restricted domain would be [-2 - sqrt(7/3), -2 + sqrt(7/3)].
B. To find the equation for the inverse function, we start by swapping the x and y variables and solving for y:
x = 3 * (y + 2) ^ 2 - 7
x + 7 = 3 * (y + 2) ^ 2
(y + 2) ^ 2 = (x + 7) / 3
y + 2 = ±sqrt((x + 7) / 3)
y = -2 ± sqrt((x + 7) / 3)
To show "half" of the graph, we need to choose one of the two possible values for y. Since we want the inverse function to pass the vertical line test, we choose the positive square root. So the equation for the "half" graph of the inverse function is:
y = -2 + sqrt((x + 7) / 3)
C. The domain of the inverse function is the range of the original function, which is [0, ∞). The range of the inverse function is the domain of the original function, which is (-∞, ∞).