Determine whether each function graphed or defined is one-to-one. y = 5|x+2|

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. Examine the given function: \(y = 5|x + 2|\). The absolute value function \(|x + 2|\) creates a V-shaped graph, which is symmetric about the vertical line \(x = -2\).
3. Because of this symmetry, for any positive value of \(y\), there are two different \(x\) values (one on each side of \(x = -2\)) that produce the same output. This violates the one-to-one condition.
4. To confirm, consider specific values: for example, \(x = -3\) and \(x = -1\) both yield \(y = 5|(-3) + 2| = 5| -1| = 5\) and \(y = 5|(-1) + 2| = 5|1| = 5\), showing two inputs with the same output.
5. Therefore, based on the shape and this test, conclude that the function \(y = 5|x + 2|\) is not one-to-one.

Key Concepts

One-to-One Function

A one-to-one function assigns each input exactly one unique output, and no two different inputs share the same output. This means the function passes the Horizontal Line Test, where any horizontal line intersects the graph at most once.

Absolute Value Function

An absolute value function outputs the distance of a number from zero, always producing non-negative values. Its graph is V-shaped and symmetric about a vertical line, which often causes it to fail the one-to-one condition because different inputs can yield the same output.

Horizontal Line Test

The Horizontal Line Test is a visual method to determine if a function is one-to-one. If any horizontal line crosses the graph more than once, the function is not one-to-one, indicating multiple inputs map to the same output.