Which of the following functions is a one-to-one function?

A one-to-one function is defined as a function where each output is associated with exactly one input. This means that for any two different inputs, the outputs must also be different.

The function represented by the logarithm, y = log x, is a one-to-one function because its graph is strictly increasing for its entire domain (x > 0). As x increases, log x also increases, ensuring that no two different x-values produce the same y-value. This property is essential for a function to be classified as one-to-one.

In contrast, the function y = x^2 does not meet this criterion because both positive and negative values of x yield the same output: for example, both 2 and -2 result in an output of 4.

The functions y = x + 3 and y = 2x are both one-to-one as well; however, the choice specified is particularly noteworthy for its properties in the context of logarithmic growth. It is important to remember that while there may be other functions that are one-to-one, the logarithmic function's unique characteristics in its defined domain provide significant relevance in many mathematical discussions, particularly in calculus and analysis.