A function f(x) is one-to-one if x1 ≠ x2 implies
f(x1) ≠ f(x2)
f(x1) ≠ f(x2). In other words, if every input x has a unique output f(x), the function is one-to-one. This means that there are no two different inputs that produce the same output.
One-to-one functions are also known as injections because they inject each input into a unique output. These types of functions are important in mathematical fields like cryptography and data compression because they allow you to store and transmit information more efficiently.
To determine if a given function is one-to-one, one can use the horizontal line test. If every horizontal line drawn through the graph of the function intersects the graph at most once, then the function is one-to-one. Alternatively, one can also use algebraic techniques such as solving for x in terms of f(x) or checking if the function passes the horizontal line test by using calculus.
Examples of one-to-one functions include linear functions, quadratic functions where a > 0, and exponential functions with positive bases. Functions that are not one-to-one include trigonometric functions such as sine and cosine, and polynomial functions with even degree where the leading coefficient is positive.
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