Linear function (calculus): Difference between revisions

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{{confuse|linear map}}

In [[calculus]] and related areas of mathematics, a '''linear function''' from the real numbers to the real numbers is a function whose graph is a [[line (geometry)|line]] in the plane.<ref>Stewart 2012, p. 23</ref>

== Properties ==

[[Image:wiki_linear_function.png|thumb|right|Graph of the linear function: {{math|1=''y''(''x'') = −''x'' + 2}}]]<!-- people, find an SVG image please instead of this abomination -->

In [[calculus]] and related areas of mathematics, a '''linear function''' from the real numbers to the real numbers is a function whose graph is a [[line (geometry)|line]] in the plane.<ref>Stewart 2012, p. 23</ref> Their characteristic property that when the value of the input variable is changed, the change in the output is a constant multiple, known as the [[slope (mathematics)|slope]], of the change in the input variable. Linear functions can be obtained from [[linear equation]]s, which are also closely related with lines.

== Properties ==

A linear function is a [[polynomial function]] in which the variable {{mvar|x}} has degree at most one, which means it is of the form

:{{math|1=f(x)=ax+b}}.<ref>Stewart 2012, p. 24</ref>

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:{{math|1=Ax+By=C}}.</math>

has {{math|B &ne; 0}}, then it may be solved for the variable {{mvar|y}} and thus used to define a linear function, namely, {{math|1=y = &minus;({{sfrac|A|B}})x + ({{sfrac|C|B}}) = f(x)}}. While all lines have equations in the general form, only the non-vertical lines have equations which can give rise to linear functions.

== Relationship with other classes of functions ==

Linear functions are a particular kind of [[polynomial function]].

Linear functions are also related to [[Exponential growth|exponential functions]]. With linear functions, increasing the input by one unit causes the output to increase by a fixed amount, which is the slope of the graph of the function. With exponential functions, increasing the input by one unit causes the output to increase by a fixed multiple, which is known as the base of the exponential function.

== Notes ==