The **sigmoid function** is a mathematical concept that has widespread uses in fields like machine learning, deep learning, and data sciences, owing to its unique properties of normalizing the output within a range of 0-1. The sigmoid function is crucial particularly in logistic regression and artificial neural networks, where it helps provide probabilities, aiding highly accurate predictions.

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## Solving the Problem with the Sigmoid Function

Understanding the problem-solving prowess of the sigmoid function demands a clear comprehension of its unique curve, which exhibits an ‘S’ shape. This curve is known as the sigmoid curve and can effectively compute real number inputs into a range between 0 and 1. This characteristic makes it a handy tool in calculating probabilities. As such, this unique property allows the sigmoid function to redefine problems, particularly where the outputs might be highly variable, by restricting them to the range of 0 and 1.

function sigmoid($t){ return 1/(1+exp(-$t)); }

The PHP function above demonstrates the implementation of the sigmoid function. The `exp()` function in PHP returns Euler’s number raised on the power of the input. Here, the input value is negated (`-t`) to ensure proper mapping of the data. The final output is then normalized by adding 1 to it.

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## Explaining the Code: Step-by-step

The sigmoid function code in PHP is quite straightforward:

function sigmoid($x){ return 1/(1+exp(-$x)); }

When calling this function, pass the desired input as the parameter of the sigmoid function. The execution of `exp(-$x)` computes the exponential representation of the negated input value. The result is then summed with 1, `(1+exp(-$x))`, providing the denominator for the fraction. Ultimately, the final step, `1/(1+exp(-$x))`, gives the output of the sigmoid function.

It’s interesting to note that the input can be any real number, and the output will always restrict between 0 and 1 — a property that makes the sigmoid function a prevalent choice in probability computations and logistic regression models.

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## Functions and Libraries Involvement

Primarily, PHP built-in function `exp()` is used in the code above. This function is a part of the **PHP Math library**, aimed to perform math operations.

- PHP Math Functions: PHP provides a wide series of mathematical functions. It does not require any installation, and we can use it directly. In the given code, the `exp()` function comes from this library.

Getting a grasp of the sigmoid function and its implementations in PHP can be fruitful for several applications, especially when handling tasks related to machine learning or data classification. Its unique ability to redefine outputs within a normalized range is what drives its extensive usage across various domains.