Understanding One-to-One Functions: Explained with Examples

One-to-One

In mathematics, a function is said to be one-to-one, or injective, if each element in the domain maps to a unique element in the codomain

In mathematics, a function is said to be one-to-one, or injective, if each element in the domain maps to a unique element in the codomain. In simpler terms, this means that no two elements in the domain of the function map to the same element in the codomain.

To determine if a function is one-to-one, we need to check if different inputs (domain elements) produce different outputs (codomain elements).

One way to determine if a function is one-to-one is by checking the horizontal line test. Draw horizontal lines across the graph of the function, and if no two lines intersect the graph at the same point, then the function is one-to-one.

Another way to check for one-to-one is by examining the function’s equation. If the equation of the function passes the horizontal line test, then it is one-to-one.

For example, consider the function f(x) = 2x. We can check if this function is one-to-one by examining its equation. If we have two different values of x, let’s say a and b, and we substitute them into the equation to get f(a) = f(b), then the function is not one-to-one. On the other hand, if we determine that f(a) ≠ f(b), then the function is one-to-one.

Let’s suppose a = 2 and b = 3. Substituting these values into the equation, we get f(2) = 2(2) = 4 and f(3) = 2(3) = 6. Since 4 ≠ 6, we can conclude that the function f(x) = 2x is one-to-one.

It is important to note that while one-to-one functions do not allow for two different inputs to map to the same output, they can have multiple outputs mapping to different inputs. In other words, a one-to-one function can have an inverse function, which allows us to recover the original input from the output.

I hope this explanation helps you understand what a one-to-one function is and how to determine if a given function is one-to-one. If you have any further questions, feel free to ask!

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