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Properties

A linear function is a polynomial function in which the variable x has degree at most one:

Such a function is called linear because its graph, the set of all points (x, f (x)) {\displaystyle (x,f(x))} in the Cartesian plane, is a line. The coefficient a is called the slope of the function and of the line (see below).

If the slope is a = 0 {\displaystyle a=0}, this is a constant function f (x) = b {\displaystyle f(x)=b} defining a horizontal line, which some authors exclude from the class of linear functions. With this definition, the degree of a linear polynomial would be exactly one, and its graph would be a line that is neither vertical nor horizontal. However, in this article, a ≠ 0 {\displaystyle a\neq 0} is required, so constant functions will be considered linear.

If b = 0 {\displaystyle b=0} then the linear function is said to be homogeneous. Such function defines a line that passes through the origin of the coordinate system, that is, the point (x, y) = (0, 0) {\displaystyle (x,y)=(0,0)}. In advanced mathematics texts, the term linear function often denotes specifically homogeneous linear functions, while the term affine function is used for the general case, which includes b ≠ 0 {\displaystyle b\neq 0}.

The natural domain of a linear function f (x) {\displaystyle f(x)}, the set of allowed input values for x, is the entire set of real numbers, x ∈ R. {\displaystyle x\in \mathbb {R}.} One can also consider such functions with x in an arbitrary field, taking the coefficients a, b in that field.

The graph y = f (x) = a x + b {\displaystyle y=f(x)=ax+b} is a non-vertical line having exactly one intersection with the y -axis, its y -intercept point (x, y) = (0, b). {\displaystyle (x,y)=(0,b).} The y -intercept value y = f (0) = b {\displaystyle y=f(0)=b} is also called the initial value of f (x). {\displaystyle f(x).} If a ≠ 0, {\displaystyle a\neq 0,} the graph is a non-horizontal line having exactly one intersection with the x -axis, the x -intercept point (x, y) = (− b a, 0). {\displaystyle (x,y)=(-{\tfrac {b}{a}},0).} The x -intercept value x = − b a, {\displaystyle x=-{\tfrac {b}{a}},} the solution of the equation f (x) = 0, {\displaystyle f(x)=0,} is also called the root or zero of f (x). {\displaystyle f(x).}