Linear function (calculus): Difference between revisions

Revision as of 03:15, 21 June 2013

Revision as of 10:16, 22 June 2013

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[[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 -->

A linear function is a [[polynomial function]] functionin withwhich one independentthe variable {{mvar|x}} ofhas degree at most one, i.e. <math>f(x)=ax+b</math>.<ref>Stewart 2012, p. 24</ref> Here {{mvar|x}} is the independent variable. The [[graph of a function|graph]] of a linear function, that is, the set of all points whose coordinates have the form ({{mvar|x}}, {{mvar|f(x)}}), is a line, which is why this type of [[Function (mathematics)|function]] is called ''linear''. Some authors, for various reasons, also require that the coefficient of the variable (the {{mvar|a}} in {{mvar|ax + b}}) should not be zero.<ref>{{harvnb|Swokowski|1983|loc=pg. 34}} is but one of many well known references that could be cited.</ref> This requirement can also be expressed by saying that the degree of the polynomial indefining this polynomialthe 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, <math>f(x) = b</math>, will be considered to be linear functions (their graphs are horizontal lines).

The [[Domain_of_a_function|___domain]] or set of allowed values for {{mvar|x}} of a linear function is the entire set of [[real number]]s {{math|'''R'''}}. This means that any real number can be substituted for {{mvar|x}}.

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The slope-intercept form of a linear equation is an equation in two variables of the form

:<math> y=ax+b </math>.

TheIn the slope-intercept form has, the variablesvariable is {{mvar|x}}, the ''independent'' variable, and {{mvar|y}}, is the ''dependent''value variableof the function. It also has two coefficients, {{mvar|a}} and {{mvar|b}}. In this instance, the fact that the values of {{mvar|y}} depend on the values of {{mvar|x}} is an expression of the functional relationship between them. To be very explicit, the linear equation is expressing the equality of values of the dependent variable {{mvar|y}} with the functional values of the linear function <math>f(x) = ax + b</math>, in other words <math>y = f(x)</math> for this particular linear function {{mvar|f}}.

If the linear function {{mvar|f}} is given, the linear equation of the graph of this function is obtained by ''defining'' the variable {{mvar|y}} to be the functional value {{mvar|f(x)}}, that is, setting <math>y = f(x) = ax + b</math> and suppressing the functional notation in the middle. Starting with a linear equation, one can create linear functions, but this is a more subtle operation and must be done with care. Why this is so is not immediately apparent when the linear equation has the slope-intercept form, so this discussion will be postponed. For the moment observe that if the linear equation has the slope-intercept form, then the expression that the dependent variable {{mvar|y}} is equal to is the linear function whose graph is the line satisfying the linear equation.