Use the horizontal line test (Theorem 2 ) to determine which functions are one-to-one. f(x)=(x^2

step 1 And graph using a graphing utility, we can see that we're going to have an open dot at x equals

Use the horizontal line test (Theorem 2 ) to determine which...

Key Concepts

One-to-One Function

A one-to-one function (or injective function) is one in which every output is produced by at most one input. This means that if f(a) = f(b) then a must equal b. This property is crucial for functions that have inverses, since only one-to-one functions can be inverted over their entire domain.

Horizontal Line Test

The horizontal line test is a graphical method used to determine if a function is one-to-one. By drawing or imagining horizontal lines across the graph of the function, one can see that if any horizontal line intersects the graph more than once, the function is not one-to-one. This test provides an intuitive visual check for injectivity.

Handling Absolute Value and Piecewise Definitions

The absolute value function often requires splitting an expression into different cases based on the sign of the input. In the context of functions, this leads to piecewise-defined functions, each of which might need to be analyzed separately to determine properties like injectivity. Understanding how absolute value transforms expressions is key to correctly analyzing such functions.

Domain Considerations in Function Analysis

When analyzing functions, especially rational functions, it is important to consider their domains. Restrictions, such as values that make the denominator zero, may affect the overall behavior and the one-to-one nature of the function. A clear understanding of the domain is necessary to accurately apply tests like the horizontal line test and to determine the existence of inverse functions.

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