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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.
If y=c is a horizontal asymptote of f(x), then y=c+k is a horizontal asymptote of f(x)+k; If a known function has an asymptote, then the scaling of the function also have an asymptote. If y=ax+b is an asymptote of f(x), then y=cax+cb is an asymptote of cf(x) For example, f(x)=e x-1 +2 has horizontal asymptote y=0+2=2, and no vertical or oblique ...
The functions whose graph is a line are generally called linear functions in the context of calculus. However, in linear algebra, a linear function is a function that maps a sum to the sum of the images of the summands. So, for this definition, the above function is linear only when c = 0, that is when the
The coefficient is −5, the indeterminates are x and y, the degree of x is two, while the degree of y is one. The degree of the entire term is the sum of the degrees of each indeterminate in it, so in this example the degree is 2 + 1 = 3. Forming a sum of several terms produces a polynomial.
The y-intercept is the initial value = = at =. The slope a measures the rate of change of the output y per unit change in the input x. In the graph, moving one unit to the right (increasing x by 1) moves the y-value up by a: that is, (+) = +.
The function () = has ″ = >, so f is a convex function. It is also strongly convex (and hence strictly convex too), with strong convexity constant 2. The function () = has ″ =, so f is a convex function. It is strictly convex, even though the second derivative is not strictly positive at all points.
In mathematics, the term linear function refers to two distinct but related notions: [1] In calculus and related areas, a linear function is a function whose graph is a straight line, that is, a polynomial function of degree zero or one. [2] For distinguishing such a linear function from the other concept, the term affine function is often used ...
If x and y are integers, rationals, or real numbers, then xy = 0 implies x = 0 or y = 0. Consider abc = 0. Then, substituting a for x and bc for y, we learn a = 0 or bc = 0. Then we can substitute again, letting x = b and y = c, to show that if bc = 0 then b = 0 or c = 0. Therefore, if abc = 0, then a = 0 or (b = 0 or c = 0), so abc = 0 implies ...