Determine whether each function graphed or defined is one-to-one. y = 2x³ - 1

Verified step by step guidance

1. Recall that a function is one-to-one if each output value corresponds to exactly one input value. This means the function passes the Horizontal Line Test: no horizontal line intersects the graph more than once.
2. Given the function \(y = 2x^3 - 1\), recognize that it is a cubic function, which generally has the form \(y = ax^3 + bx^2 + cx + d\).
3. To determine if the function is one-to-one, analyze its derivative to check if the function is strictly increasing or strictly decreasing. Compute the derivative: \(y' = \frac{d}{dx}(2x^3 - 1) = 6x^2\).
4. Since \(6x^2 \geq 0\) for all real \(x\) and equals zero only at \(x=0\), the function is non-decreasing everywhere and strictly increasing except possibly at one point. This suggests the function is one-to-one.
5. Conclude that because the function is strictly increasing (except at a single point where the slope is zero but does not decrease), it passes the Horizontal Line Test and is therefore 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.

Cubic Functions

Cubic functions are polynomial functions of degree three, typically in the form y = ax^3 + bx^2 + cx + d. They often have an S-shaped curve and can be strictly increasing or decreasing, which affects whether they are one-to-one.

Horizontal Line Test

The horizontal line test is a graphical method to determine if a function is one-to-one. If any horizontal line intersects the graph more than once, the function is not one-to-one; otherwise, it is.