i) To find the values of a and b, we need to make use of the given information about the function fg(x).
Given that fg(x) = 6x^2 - 21 for x ≤ q, we can substitute x = q into fg(x) and equate it to g(x) to form an equation:
6q^2 - 21 = aq^2 + bq
From here, we have one equation with two unknowns (a and b). To solve for a and b, we need another equation involving a and b. We can make use of the information about the function f(x).
Given that f(x) = 2x + 3 for x ≥ 0, we can substitute x = 0 into f(x) and equate it to g(x) for x = q:
2(0) + 3 = a(0)^2 + bq
3 = bq
Now we have two equations:
1) 6q^2 - 21 = aq^2 + bq
2) 3 = bq
Solving these two equations simultaneously will give us the values of a and b.
ii) To find the greatest possible value of q, given that q = -3, we need to consider the domain of the function fg(x). The function fg(x) is defined as fg(x) = 6x^2 - 21 for x ≤ q. In this case, x ≤ q means that x cannot be greater than q. Therefore, the greatest possible value of q is the largest value that x can take while still satisfying the condition x ≤ q. In this case, q = -3, so the greatest possible value of q is -3.
iii) To find the range of fg, we need to determine the set of all possible y-values that fg(x) can take. From the equation fg(x) = 6x^2 - 21, we can observe that the coefficient of the x^2 term is positive (6), which means that the parabola opens upward. This implies that fg(x) can take on any value greater than or equal to the vertex of the parabola. The vertex of the parabola can be determined by finding the x-coordinate of the vertex using the formula x = -b/2a.
For fg(x) = 6x^2 - 21, a = 6 and b = 0. Substituting these values into the formula x = -b/2a, we get:
x = -(0)/(2(6)) = 0
Therefore, the vertex of the parabola is (0, -21).
Since the parabola opens upward and the vertex is the lowest point on the graph, the range of fg is (-21, ∞).
iv) To find the expression for the inverse of (fg)(x) and state its domain, we need to switch the roles of x and y in the equation fg(x) = 6x^2 - 21 and solve for x.
Start with fg(x) = 6x^2 - 21:
y = 6x^2 - 21
Switching x and y:
x = 6y^2 - 21
Now solve for y:
6y^2 = x + 21
y^2 = (x + 21)/6
y = ±sqrt((x + 21)/6)
The expression for the inverse of (fg)(x) is y = ±sqrt((x + 21)/6).
For the domain of the inverse of (fg), we need to consider the values of x that make the square root expression defined. Since the square root function is only defined for non-negative values, the expression inside the square root, (x + 21)/6, must be greater than or equal to 0:
(x + 21)/6 ≥ 0
Solving this inequality:
x + 21 ≥ 0
x ≥ -21
Therefore, the domain of the inverse of (fg) is x ≥ -21.