Determine whether each function graphed or defined is one-to-one. y = 2x^3 - 1

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 no horizontal line intersects the graph of the function more than once.
2. Consider the function given: \( y = 2x^3 - 1 \). This is a cubic function, which is generally a type of polynomial function.
3. Apply the Horizontal Line Test: For a function to be one-to-one, any horizontal line should intersect the graph at most once.
4. Analyze the behavior of the function: Since \( y = 2x^3 - 1 \) is a cubic function, it is continuous and has no repeating values for different inputs, suggesting it passes the Horizontal Line Test.
5. Conclude based on the analysis: Since the function passes the Horizontal Line Test, it is 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 visually assessed 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 Functions

Cubic functions are polynomial functions of degree three, typically expressed in the form y = ax^3 + bx^2 + cx + d. These functions can exhibit various behaviors, including having one or two turning points, which can affect their one-to-one nature. The specific function given, y = 2x^3 - 1, is a cubic function that generally increases without bound.

Graphical Analysis

Graphical analysis involves examining the shape and behavior of a function's graph to derive insights about its properties. For the function y = 2x^3 - 1, analyzing its graph can help determine if it is one-to-one by observing whether any horizontal lines intersect the graph more than once, which would indicate that it is not one-to-one.