Final answer:
To find the range of the function f(x) = 3x - x², we analyze the behavior of the function and determine the maximum value at the vertex.
Step-by-step explanation:
To find the range of the function f(x) = 3x - x², we need to determine the set of all possible output values. In order to do this, we can analyze the behavior of the function.
The function is a quadratic equation, which means it has a parabolic shape. Since the coefficient of the x² term is negative, the parabola opens downwards. This tells us that the maximum value of the function occurs at the vertex.
The x-coordinate of the vertex can be found using the formula x = -b/2a, where a and b are the coefficients of the x² and x terms, respectively. In this case, a = -1 and b = 3, so the x-coordinate of the vertex is x = -3/(2*(-1)) = 3/2 = 1.5.
Now we can substitute this x-value into the function to find the corresponding y-value: f(1.5) = 3(1.5) - (1.5)² = 4.5 - 2.25 = 2.25.
Therefore, the range of the function f(x) = 3x - x² is (-∞, 2.25].