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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 with uniform scales) is a line in the plane.〔Stewart 2012, p. 23〕 Their characteristic property that when the value of the input variable is changed, the change in the output is a constant multiple of the change in the input variable. Linear functions are related to linear equations. == Properties == A linear function is a polynomial function in which the variable has degree at most one, which means it is of the form〔Stewart 2012, p. 24〕 :. Here is the variable. The graph of a linear function, that is, the set of all points whose coordinates have the form , is a line on the Cartesian plane (if over real numbers). That is why this type of function is called ''linear''. Some authors, for various reasons, also require that the coefficient of the variable (the in ) should not be zero.〔 is but one of many well known references that could be cited.〕 This requirement can also be expressed by saying that the degree of the polynomial defining the function is exactly one, or by saying that the line which is the graph of a linear function is a ''slanted'' line (neither vertical nor horizontal). This requirement will not be imposed in this article, thus constant functions, , will be considered to be linear functions (their graphs are horizontal lines). The domain or set of allowed values for of a linear function is the entire set of real numbers , or whatever field that is in use. This means that any (real) number can be substituted for . Because two different points determine a line, it is enough to substitute two different values for in the linear function and determine for each of these values. This will give the coordinates of two different points that lie on the line. Because is a function, this line will not be vertical. If the value of either or both of the coefficient letters and are changed, a different line is obtained. Since the graph of a linear function is a nonvertical line, this line has exactly one intersection point with the -axis. This point is . The graph of a nonconstant linear function has exactly one intersection point with the -axis. This point is . From this, it follows that a nonconstant linear function has exactly one zero or root. That is, there is exactly one solution to the equation . The zero is . 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Linear function (calculus)」の詳細全文を読む スポンサード リンク
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