Determine whether each function graphed or defined is one-to-one. y = ∛(x+1) - 3

Verified step by step guidance

1. Recall that a function is one-to-one if each output corresponds to exactly one input. This means the function passes the Horizontal Line Test: no horizontal line intersects the graph more than once.
2. Identify the given function: \(y = \sqrt[3]{x + 1} - 3\). This is a cube root function shifted horizontally and vertically.
3. Understand the behavior of the cube root function \(y = \sqrt[3]{x}\). It is an increasing function over all real numbers, meaning it is one-to-one because it never repeats the same output for different inputs.
4. Since the function \(y = \sqrt[3]{x + 1} - 3\) is a horizontal shift by \(-1\) and a vertical shift by \(-3\) of the cube root function, these transformations do not affect the one-to-one nature of the function.
5. Conclude that because the base cube root function is one-to-one and shifts do not change this property, the given function \(y = \sqrt[3]{x + 1} - 3\) is also one-to-one.

Key Concepts

One-to-One Function

A function is one-to-one if each output corresponds to exactly one input, meaning no two different inputs produce the same output. This property ensures the function has an inverse that is also a function.

Cube Root Function

The cube root function, y = ∛x, is defined for all real numbers and is strictly increasing, which means it passes the horizontal line test and is one-to-one. Understanding its shape helps determine the one-to-one nature of transformations.

Function Transformations

Transformations such as shifts and translations (e.g., y = ∛x + 1 - 3) move the graph horizontally or vertically without changing its shape. These transformations do not affect the one-to-one property of the original function.