Final answer:
The correct representation of the function f(x) = (x-3)² is a parabola with its vertex at (3,0) extending upwards. The range of this function is [0, ∞) since it opens upwards and the lowest value of f(x) is 0.
Step-by-step explanation:
To find the correct representation of the function f(x) = (x-3)² and state its range, we first need to understand the form of the function given. This function is in the form of a squared binomial, which represents a parabola that opens upwards with its vertex at (3,0). The range of a function is the set of all possible output values (y-values), which depends on how the graph of the function looks.
Since the parabola opens upwards and has its vertex at the point (3,0), the smallest value that f(x) can take is 0. This occurs when x is equal to 3. As x moves away from 3, the value of f(x) increases because the function is squared. Therefore, the range of the function is [0, ∞), as the value of f(x) starts at 0 and increases to infinity as x moves away from 3 in either direction.
The correct representation of the given function would show a parabola with its vertex at (3,0) and extending upwards from that point. On the coordinate system, we would label the f(x) on the y-axis and x on the x-axis, with the parabola properly scaled to reflect the behaviour of the function across the domain.