One-to-one Function: Explanation

Definition: A function f is said to be one-to-one if and only if f(a) = f(b) implies that a = b for all a and b in the domain of f.

Explanation:
To understand the concept of a one-to-one function, let's break down the definition and examine each part.

1. Function: A function is a relation between two sets, where each element in the first set (called the domain) is related to exactly one element in the second set (called the codomain). In other words, for each input, there is only one output.

2. One-to-One: A function is said to be one-to-one if every output value is associated with a unique input value. It means that if two different inputs produce the same output, then the function is not one-to-one.

3. f(a) = f(b) implies that a = b: This condition states that if the outputs of the function for two different inputs are the same, then the inputs must also be the same. In other words, if f(a) = f(b), then a and b must be equal.

4. For all a and b in the domain of f: This condition applies to all possible input values a and b in the domain of the function. It means that the one-to-one property holds true for every pair of inputs.

Example:
Let's consider a simple function f(x) = x^2. This function is not one-to-one because different inputs can produce the same output. For example, f(2) = f(-2) = 4, but 2 ≠ -2.

On the other hand, if we consider the function g(x) = 2x, this function is one-to-one. If g(a) = g(b), then 2a = 2b, and by dividing both sides by 2, we get a = b. Therefore, the function g(x) = 2x satisfies the one-to-one property.

Conclusion:
In summary, a function is said to be one-to-one if and only if each output value is associated with a unique input value. If two different inputs produce the same output, then the function is not one-to-one. The condition f(a) = f(b) implies that a = b for all a and b in the domain of the function.