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Given a function: from a set X (the domain) to a set Y (the codomain), the graph of the function is the set [4] = {(, ()):}, which is a subset of the Cartesian product.In the definition of a function in terms of set theory, it is common to identify a function with its graph, although, formally, a function is formed by the triple consisting of its domain, its codomain and its graph.
Constant function: polynomial of degree zero, graph is a horizontal straight line; Linear function: First degree polynomial, graph is a straight line. Quadratic function: Second degree polynomial, graph is a parabola. Cubic function: Third degree polynomial. Quartic function: Fourth degree polynomial. Quintic function: Fifth degree polynomial.
Equivalently, a function is convex if its epigraph (the set of points on or above the graph of the function) is a convex set. In simple terms, a convex function graph is shaped like a cup ∪ {\displaystyle \cup } (or a straight line like a linear function), while a concave function 's graph is shaped like a cap ∩ {\displaystyle \cap } .
The second derivative of a function f can be used to determine the concavity of the graph of f. [2] A function whose second derivative is positive is said to be concave up (also referred to as convex), meaning that the tangent line near the point where it touches the function will lie below the graph of the function.
Graph of a linear function Graph of a polynomial function, here a quadratic function. Graph of two trigonometric functions: sine and cosine. A real function is a real-valued function of a real variable, that is, a function whose codomain is the field of real numbers and whose domain is a set of real numbers that contains an interval.
[2] [3] For example, the graph of the differentiable function has an inflection point at (x, f(x)) if and only if its first derivative f' has an isolated extremum at x. (this is not the same as saying that f has an extremum). That is, in some neighborhood, x is the one and only point at which f' has a (local) minimum or maximum.
In calculus and related areas of mathematics, a linear function from the real numbers to the real numbers is a function whose graph (in Cartesian coordinates) is a non-vertical line in the plane. [1] The characteristic property of linear functions is that when the input variable is changed, the change in the output is proportional to the change ...
the function f is n − 1 times continuously differentiable on the closed interval [a, b] and the n th derivative exists on the open interval (a, b), and; there are n intervals given by a 1 < b 1 ≤ a 2 < b 2 ≤ ⋯ ≤ a n < b n in [a, b] such that f (a k) = f (b k) for every k from 1 to n. Then there is a number c in (a, b) such that the n ...
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