Final answer:
To reflect the function f(x) = |x| over the x-axis and then translate it 4 units up and 3 units to the left, the function rule for g(x) becomes g(x) = -|x + 3| + 4. This entails inverting the graph and shifting it horizontally and vertically as per algebraic rules.
Step-by-step explanation:
The question involves transforming a basic absolute value function algebraically by reflecting it over the x-axis and then translating it. The absolute value function, denoted as f(x) = |x|, is known for its distinctive 'V' shape, with its vertex at the origin (0,0).
To reflect f(x) over the x-axis, every y-value must be multiplied by -1, resulting in g(x) = -|x|. This reflection will invert the graph, flipping the 'V' shape upside down.
Next, translating the function 4 units up and 3 units to the left involves two alterations. The translation up is achieved by adding 4 to our function, giving us g(x) = -|x| + 4. To translate the function 3 units to the left, we must replace x with (x + 3), resulting in the function g(x) = -|x + 3| + 4. This is the final transformed function.
These transformations follow basic algebra rules where horizontal shifts are represented by changes inside the absolute value (reflecting addition or subtraction of the x-value), while vertical shifts are represented by addition or subtraction outside the function.