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

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 is paired with exactly one input value, and no output value is repeated for different input values.
2. Consider the function given: \( y = -\frac{1}{x+2} \). This is a rational function.
3. To determine if the function is one-to-one, check if it passes the Horizontal Line Test: A function is one-to-one if every horizontal line intersects the graph of the function at most once.
4. Analyze the behavior of the function: The function \( y = -\frac{1}{x+2} \) is a hyperbola, which typically has two branches. However, due to the transformation, it may behave differently.
5. Graph the function or consider its algebraic properties to see if any horizontal line intersects the graph more than once. If no horizontal line does, the function is one-to-one; otherwise, it is not.

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, one 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.

Graphing Rational Functions

Rational functions are expressed as the ratio of two polynomials. The function given, y = -1/(x + 2), is a rational function that can be graphed to analyze its behavior. Understanding the asymptotes, intercepts, and overall shape of the graph is crucial for determining properties like whether it is one-to-one.

Horizontal Line Test

The horizontal line test is a visual method used to determine if a function is one-to-one. If any horizontal line drawn across the graph intersects the curve at more than one point, the function fails the test and is not one-to-one. This test is particularly useful for analyzing the graphs of functions quickly and effectively.