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Functions of the form ae x for constant a are the only functions that are equal to their derivative (by the Picard–Lindelöf theorem). Other ways of saying the same thing include: The slope of the graph at any point is the height of the function at that point. The rate of increase of the function at x is equal to the value of the function at x.
The product logarithm Lambert W function plotted in the complex plane from −2 − 2i to 2 + 2i The graph of y = W(x) for real x < 6 and y > −4.The upper branch (blue) with y ≥ −1 is the graph of the function W 0 (principal branch), the lower branch (magenta) with y ≤ −1 is the graph of the function W −1.
For example the function () = grows at an ever increasing rate, but is much slower than growing exponentially. For example, when =, it grows at 3 times its size, but when = it grows at 30% of its size. If an exponentially growing function grows at a rate that is 3 times is present size, then it always grows at a rate that is 3 times its present ...
In mathematics, for a function :, the image of an input value is the single output value produced by when passed .The preimage of an output value is the set of input values that produce .
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. [ 1 ] [ 2 ] [ 3 ] This concept first arose in calculus , and was later generalized to the more abstract setting of order theory .
In mathematics, a coercive function is a function that "grows rapidly" at the extremes of the space on which it is defined. Depending on the context different exact ...
Thomae's function: is a function that is continuous at all irrational numbers and discontinuous at all rational numbers. It is also a modification of Dirichlet function and sometimes called Riemann function. Kronecker delta function: is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise.
With exponential functions, increasing the input by one unit causes the output to increase by a fixed multiple, which is known as the base of the exponential function. If both arguments and values of a function are in the logarithmic scale (i.e., when log(y) is a linear function of log(x)), then the straight line represents a power law: