Linear function (calculus)

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. 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.

A linear function is a polynomial function in which the variable x has degree at most one, which means it is of the form

Here x is the variable. The graph of a linear function, that is, the set of all points whose coordinates have the form (x, f(x)), 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 a in ax + b) should not be zero. 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, f(x) = b, will be considered to be linear functions (their graphs are horizontal lines).

The domain or set of allowed values for x of a linear function is the entire set of real numbers R, or whatever field that is in use. This means that any (real) number can be substituted for x.

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