Determine whether each function graphed or defined is one-to-one. y = 2(x+1)² - 6

Verified step by step guidance

1. Recall that a function is one-to-one if and only 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: \(y = 2(x+1)^2 - 6\). Notice this is a quadratic function in vertex form, where the squared term \((x+1)^2\) suggests a parabola.
3. Since the parabola opens upwards (because the coefficient 2 is positive), it is symmetric about the vertical line \(x = -1\). This symmetry means for some \(y\)-values, there are two different \(x\)-values producing the same output.
4. To confirm, consider the vertex at \(x = -1\). For \(y\) values greater than the vertex's \(y\)-coordinate, there will be two \(x\) values (one on each side of the vertex) that yield the same \(y\). This violates the one-to-one condition.
5. Therefore, conclude that the function \(y = 2(x+1)^2 - 6\) is not one-to-one because it fails the Horizontal Line Test due to its parabolic shape.

Key Concepts

One-to-One Function

A one-to-one function assigns each input exactly one unique output, and each output corresponds to exactly one input. This means no two different inputs produce the same output. Determining if a function is one-to-one helps understand if it has an inverse function.

Quadratic Functions and Their Graphs

Quadratic functions are polynomial functions of degree two, typically graphed as parabolas. The function y = 2(x+1)^2 - 6 is a parabola opening upwards, shifted left by 1 and down by 6. Parabolas are generally not one-to-one because they fail the horizontal line test.

Horizontal Line Test

The horizontal line test is a visual 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. For the given quadratic, horizontal lines above the vertex intersect twice, indicating it is not one-to-one.