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

Verified step by step guidance

1. Understand the definition of a one-to-one function: A function is one-to-one if each output value is paired with exactly one input value, meaning it passes the horizontal line test.
2. Consider the function given: \( y = \sqrt[3]{x+1} - 3 \). This is a transformation of the cube root function.
3. Recall that the cube root function \( y = \sqrt[3]{x} \) is one-to-one because it passes the horizontal line test.
4. Analyze the transformations: The function \( y = \sqrt[3]{x+1} - 3 \) involves a horizontal shift left by 1 unit and a vertical shift down by 3 units.
5. Conclude that these transformations do not affect the one-to-one nature of the cube root function, so \( y = \sqrt[3]{x+1} - 3 \) is also one-to-one.

Key Concepts

One-to-One Functions

A function is considered one-to-one if it assigns a unique output for every unique input, meaning no two different inputs produce the same output. This can be tested using the horizontal line test: if any horizontal line intersects the graph of the function more than once, the function is not one-to-one.

Cubic Root Function

The cubic root function, represented as y = ∛x, is a type of radical function that returns the number which, when cubed, gives the input value x. This function is defined for all real numbers and is continuous and increasing, which contributes to its one-to-one nature.

Transformations of Functions

Transformations involve shifting, reflecting, stretching, or compressing the graph of a function. In the given function y = ∛x + 1 - 3, the transformations include a vertical shift of +1 and a downward shift of -3, which do not affect the one-to-one property of the original cubic root function.