one -one function
(what are its conditions)
A one-to-one function, also known as an injective function, is a function where every element in the domain map to a distinct element in the range.
The conditions for a function to be one-to-one are as follows:
1. Every element in the domain must have a unique image in the range: This means that for every x1 and x2 in the domain, if x1 is not equal to x2, then f(x1) must not be equal to f(x2). In other words, different inputs should always produce different outputs.
2. No two different elements in the domain should have the same image in the range: This means that if f(x1) = f(x2), then it must be true that x1 is equal to x2. In other words, the same output should never be produced by different inputs.
To check if a given function is one-to-one, you can use the following method:
1. Suppose the function is given as f(x).
2. Choose two arbitrary inputs, let's say x1 and x2, from the domain.
3. Evaluate f(x1) and f(x2).
4. Compare the outputs. If f(x1) ≠ f(x2), where ≠ denotes "not equal to," then the function is one-to-one. If f(x1) = f(x2), then the function is not one-to-one.
By following these conditions and steps, you can determine if a function is one-to-one or not.