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In mathematics it is possible to have functions of functions, and functions of functions of functions and so on.So if f is a function of function g of a variable x, which may be written as f(g(x)) then g is the inner function.Thus in the function sin(3x2+5), the inner function is (3x2+5). Inner functions are particularly important in calculus (differentiation and integration).
Every function differs from every other function. Otherwise they would not be different functions!
There are infinitely many types of functions. For example: Discrete function, Continuous functions, Differentiable functions, Monotonic functions, Odd functions, Even functions, Invertible functions. Another way of classifying them gives: Logarithmic functions, Inverse functions, Algebraic functions, Trigonometric functions, Exponential functions, Hyperbolic functions.
A piecewise function is a function defined by two or more equations. A step functions is a piecewise function defined by a constant value over each part of its domain. You can write absolute value functions and step functions as piecewise functions so they're easier to graph.
An inverse is NOT called a circular function. Only inverse functions that are circular functions are called circular functions for obvious reasons.