Given the function f, prove that f is one-to-one using the definition of a one-to-one function o

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Given the function f, prove that f is one-to-one using the...

Key Concepts

Algebraic Manipulation in Proofs

Algebraic manipulation involves rearranging and simplifying equations to isolate variables and derive conclusions. In proving a function is one-to-one, it is crucial to manipulate the equality f(a) = f(b) properly, such as by adding, subtracting, or dividing by nonzero numbers, to ultimately demonstrate that a equals b.

Properties of Linear Functions

Linear functions of the form f(x) = mx + b, where m is nonzero, are inherently one-to-one because their slopes ensure a consistent rate of change. This means that as x increases or decreases, the output f(x) changes in a unique and predictable manner, reinforcing the concept that different inputs produce distinct outputs.

One-to-one Function (Injectivity)

A function is one-to-one, or injective, if and only if for any two elements in its domain, whenever the functions' outputs are equal, the inputs are equal as well. This property ensures that each element of the codomain is related to at most one element of the domain, meaning that no two different inputs can produce the same output.

Direct Proof using the Definition

To prove that a function is one-to-one, a common method is to assume that the outputs corresponding to two arbitrary inputs are equal and then show through algebraic manipulation that the inputs themselves must be equal. This direct approach uses the definition of injectivity and provides a clear logical sequence from the hypothesis f(a) = f(b) to the conclusion a = b.

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