In Exercises 107-118, begin by graphing the cube root function, f(x) = ∛x. Then use transformations of this graph to graph the given function. g(x) = ∛(x-2)

Key Concepts

Cube Root Function

The cube root function, denoted as f(x) = ∛x, is a fundamental mathematical function that returns the number which, when cubed, gives the input value x. This function is defined for all real numbers and has a characteristic S-shaped curve that passes through the origin (0,0). Understanding its basic shape and properties is essential for graphing and transforming it.

Graph Transformations

Graph transformations involve shifting, stretching, compressing, or reflecting the graph of a function. In this case, the transformation applied to the cube root function f(x) = ∛x to obtain g(x) = ∛(x-2) is a horizontal shift to the right by 2 units. Recognizing how these transformations affect the graph is crucial for accurately sketching the new function.

Horizontal Shifts

A horizontal shift occurs when a function is modified by adding or subtracting a constant from the input variable. For g(x) = ∛(x-2), the '-2' indicates a shift of the graph of f(x) = ∛x to the right by 2 units. This concept is vital for understanding how the position of the graph changes without altering its shape.