Determine whether each function graphed or defined is one-to-one.

Verified step by step guidance

1. Step 1: Understand the definition of a one-to-one function. A function is one-to-one if each output (y-value) corresponds to exactly one input (x-value). In other words, no horizontal line intersects the graph more than once.
2. Step 2: Observe the graph of the function. The graph is a straight line with a negative slope, which means it is a linear function of the form \(y = mx + b\) where \(m < 0\).
3. Step 3: Apply the Horizontal Line Test. Since the graph is a straight line with a non-zero slope, any horizontal line will intersect the graph at exactly one point.
4. Step 4: Conclude that the function passes the Horizontal Line Test, indicating it is one-to-one.
5. Step 5: Summarize that because the function is linear with a non-zero slope and passes the Horizontal Line Test, it is a one-to-one function.

Key Concepts

One-to-One Function

A function is one-to-one if each output value corresponds to exactly one input value. This means no two different inputs produce the same output. One-to-one functions have an inverse that is also a function.

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. If every horizontal line intersects at most once, the function is one-to-one.

Linear Functions and Slope

Linear functions have the form y = mx + b, where m is the slope. If the slope is nonzero, the function is one-to-one because it is strictly increasing or decreasing, ensuring unique outputs for each input.