To determine if a function is one-to-one (or injective), we need to check if different inputs produce different outputs. In mathematical terms, a function f: A → A is one-to-one if, for any two elements x1 and x2 in the domain A, the condition f(x1) = f(x2) implies that x1 = x2. This means that if two outputs are equal, their corresponding inputs must also be equal.
Let’s analyze each of the functions a, b, c, and d from A to A:
- Function A: Consider the function f(x) = 2x. It is injective because if f(x1) = f(x2), then 2x1 = 2x2. Dividing both sides by 2 yields x1 = x2.
- Function B: If we look at f(x) = x2, it is not one-to-one if considered over the entire set of real numbers since f(-1) = f(1) = 1, but -1 ≠ 1. Therefore, this function is not injective.
- Function C: Next, consider f(x) = x + 3. This function is one-to-one since for any two outputs, f(x1) = f(x2) implies x1 + 3 = x2 + 3, hence x1 = x2.
- Function D: Lastly, with f(x) = 3 – x, this is also one-to-one because f(x1) = f(x2) translates to 3 – x1 = 3 – x2 which simplifies to x1 = x2.
In summary:
- Function A: One-to-One
- Function B: Not One-to-One
- Function C: One-to-One
- Function D: One-to-One
To conclude, analyzing each function using the definition of injectivity and looking for unique outputs corresponding to unique inputs helps us determine which functions are one-to-one.