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

Verified step by step guidance

1. Identify the given function: \(y = -\frac{1}{x} + 2\).
2. Recall that a function is one-to-one if each \(y\)-value corresponds to exactly one \(x\)-value, meaning the function passes the Horizontal Line Test.
3. Analyze the function's behavior: since \(y = -\frac{1}{x} + 2\) is a transformation of the reciprocal function \(y = \frac{1}{x}\), consider how the negative sign and the +2 shift affect its graph.
4. Check if the function is strictly increasing or strictly decreasing on its domain, because strictly monotonic functions are one-to-one.
5. Conclude by determining if any horizontal line intersects the graph more than once, which would indicate the function is not 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.

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.

Rational Functions and Their Graphs

Rational functions are ratios of polynomials, often with vertical and horizontal asymptotes. Understanding their shape helps analyze behavior and determine if they pass the horizontal line test for one-to-one properties.