Final answer:
The domain of the function f(x) = 6x² + 9 is all real numbers, and its range is y ≥ 9 because it is a quadratic function with a positive leading coefficient and the vertex (the minimum point) has a y-coordinate of 9.
Step-by-step explanation:
The domain of a function refers to the set of all possible input values (x-values) for which the function is defined, while the range refers to the set of all possible output values (y-values). For the function f(x) = 6x² + 9, since it is a quadratic function with a positive leading coefficient, it opens upwards and the vertex represents the minimum point on the graph.
The domain of this function is all real numbers because there are no restrictions on the values that x can take for a quadratic function. Hence, the domain is x ∈ R, which means x belongs to the set of real numbers.
The range of this function starts from the y-coordinate of the vertex and goes to infinity. The vertex occurs at x = 0, so the minimum y-value is f(0) = 6(0)² + 9 = 9. Thus, the range of this function is y ≥ 9, meaning y is greater than or equal to 9.
Therefore, the correct answer is c) Domain: x ∈ R, Range: y ≥ 9.