Answer:
Explanation:
The graph of the function g(x) = 6x - x² can be determined by analyzing its equation. This equation represents a quadratic function, which means the graph will be a parabola. To find the vertex of the parabola, we can use the formula x = -b/(2a), where a and b are the coefficients of the quadratic function. In this case, a = -1 and b = 6. Substituting these values into the formula, we have x = -6/(2*(-1)) = -6/(-2) = 3. So, the x-coordinate of the vertex is 3. To find the corresponding y-coordinate, we substitute x = 3 into the equation g(x) = 6x - x²: g(3) = 6(3) - (3)² = 18 - 9 = 9. Therefore, the vertex of the parabola is (3, 9). Since the coefficient of the x² term is negative, the parabola opens downwards. This means the graph of g(x) is a concave downward parabola. The domain of h(x) is determined by the domain of the composition of functions f(g(x)). In this case, f(x) = √x, which has a domain of x ≥ 0, since the square root function is only defined for non-negative real numbers. However, the domain of g(x) is determined by the quadratic term, which in this case is -x². A quadratic function is defined for all real numbers, so the domain of g(x) is (-∞, ∞). Since the composition of functions h(x) = f(g(x)) involves both f(x) and g(x), the domain of h(x) will be determined by the common domain of f(x) and g(x). In this case, the common domain is the intersection of the domain of f(x) (x ≥ 0) and the domain of g(x) (-∞, ∞).