Final answer:
The function g(x) = -f(x + 3) represents a leftward translation by 3 units and a vertical reflection across the x-axis. This translates 'left' and flips 'upside-down' the graph of f(x) to obtain g(x).
Step-by-step explanation:
When analyzing the transformation of the function f(x) to get g(x), it is evident that there are two transformations occurring due to the new rule g(x) = -f(x + 3).
Firstly, the addition of 3 inside the function argument, x + 3, indicates a translation to the left by 3 units. Algebraic expressions such as f(x - d) move the graph to the right by d units, whereas in this case, f(x + d) translates it to the left by d units. This moves the entire graph of f(x) horizontally on the coordinate plane.
Secondly, the negative sign before f(x) introduces a vertical reflection about the x-axis. A function that undergoes -f(x) is reflected in such a way that if the original function had a positive output at any given x, the new function outputs a negative of that value, effectively flipping the graph upside down.
A big explanation for this is that combining a horizontal translation and a vertical reflection defines the new positions of all points of the original graph f(x) within the coordinate plane. This creates the transformed function g(x).