Which function outputs a value between 0 and 1, forming an S shape in logistic regression?

The logistic function is the correct choice as it uniquely outputs values that range between 0 and 1, creating an S-shaped curve known as the sigmoid curve. This property makes it particularly useful in logistic regression, as it effectively models the probability of a binary outcome. The shape of the function allows it to smoothly transition from 0 to 1, making it ideal for scenarios where we want to predict probabilities that are bounded within this range.

The logistic function is mathematically defined as:

[ f(x) = \frac{1}{1 + e^{-x}} ]

where ( e ) is the base of the natural logarithm. As ( x ) approaches negative infinity, the function approaches 0, and as ( x ) increases toward positive infinity, the function approaches 1, thus demonstrating its S-shaped characteristic.

In contrast, the linear function produces a straight line and does not constrain its output between 0 and 1. The sine function produces oscillating values, while the exponential function grows or decays rapidly, neither of which fits the need for a bounded probability output. Therefore, the logistic function is the best fit for producing the desired output in logistic regression.