Determine whether each function graphed or defined is one-to-one. y = 2x - 8

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 corresponds to exactly one input value, and no two different input values map to the same output value.
2. Use the Horizontal Line Test: A function is one-to-one if and only if no horizontal line intersects the graph of the function more than once.
3. Consider the function given: \( y = 2x - 8 \). This is a linear function, which graphs as a straight line.
4. Apply the Horizontal Line Test to the graph of \( y = 2x - 8 \): Since a straight line with a non-zero slope will never be horizontal, any horizontal line will intersect it at most once.
5. Conclude that since the graph of \( y = 2x - 8 \) passes the Horizontal Line Test, the function is one-to-one.

Key Concepts

One-to-One Function

A one-to-one function is a type of function where each output value is associated with exactly one input value. This means that no two different inputs produce the same output. To determine if a function is one-to-one, you can use the horizontal line test: if any horizontal line intersects the graph of the function more than once, the function is not one-to-one.

Linear Functions

Linear functions are mathematical expressions that create a straight line when graphed. They can be represented in the form y = mx + b, where m is the slope and b is the y-intercept. The function given, y = 2x - 8, is a linear function with a slope of 2 and a y-intercept of -8, which indicates that it will produce a straight line on a graph.

Horizontal Line Test

The horizontal line test is a method used to determine if a function is one-to-one. If any horizontal line drawn across the graph of the function intersects it at more than one point, the function fails the test and is not one-to-one. For linear functions like y = 2x - 8, which are straight lines, they will always pass the horizontal line test, confirming they are one-to-one.