Answer:
f(x) = (x + 1)³ - (x + 1)² = x³ + 2 x² + x
Explanation:
To shift g(x) to the left by one unit, replace all x in the expression of g(x) with (x + 1). How does this work? For example, g(x) = -2 for x = -1 in this question. f(x) = g(x + 1) may also equate -2, but only one unit to the left of x = -1 at x = -2 since -2 + 1 = -1. So is the case for the rest of the points on g(x). f(x) = g(x + 1) leads to the same set of values but only at one unit to the left. Adding one to the independent variable of g(x) shifts the graph to the left by one unit.
To find g(x + 1), replace every x in the equation x³ - x² with (x + 1):
f(x) = g(x + 1) = (x + 1)³ - (x + 1)².
Expand (x + 1)³ - (x + 1)² using the binomial theorem:
f(x) = (x + 1)³ - (x + 1)² = x³ + 3 x² + 3 x + 1 - (x² + 2 x + 1) = x³ + 2 x² + x