Final answer:
The transformation that maps the cubic function f(x) to the quadratic function g(x) involves a horizontal shift of 2 units to the right. However, they cannot be completely transformed into each other as f(x) is cubic and g(x) is quadratic.
Step-by-step explanation:
To describe the transformation that maps f(x) = (x+1)x² onto g(x) = (x+3)², we need to analyze the changes in structure between these two functions. The function f(x) is a cubic function, specifically the product of a linear term (x+1) and a quadratic term (x²), while g(x) is a quadratic function with a squared linear term (x+3).²
First, notice that if we extract the x from the cubic term of f(x), we would have x*(x+1)². This resembles g(x) in that (x+1)² is similar to (x+3)², except for the different constants inside the brackets. This indicates a horizontal shift. Since x+3 is equivalent to (x+1)+2, we can deduce that a shift of 2 units to the right has been applied to transform (x+1)² into (x+3)².
However, the presence of the x term multiplying the squared term in f(x) turns what would be a straightforward quadratic into a cubic function. To map the cubic function f(x) onto the quadratic g(x), one must 'remove' this linear x factor. This cannot be accomplished through standard function transformations such as shifts, stretches, or reflections commonly discussed in the context of function transformation.
In summary, there is a horizontal shift of 2 units to the right involved, but due to the fundamental difference in the degree of the two polynomials - cubic for f(x) and quadratic for g(x) - they cannot be transformed into each other through the basic transformations alone.