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. The characteristic property of linear functions is that when the input variable is changed, the change in the output is proportional to the change in the input. 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. The characteristic property of linear functions is that when the input variable is changed, the change in the output is proportional to the change in the input. Linear functions are related to linear equations. 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, its graph a diagonal line neither vertical nor horizontal. However, we will not require a ≠ 0 {displaystyle a eq 0} in this article, so constant functions will be considered linear. 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 xin 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 eq 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)=(-{ frac {b}{a}},0)} . The x-intercept value x = − b a {displaystyle x=-{ frac {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)} . The slope of a nonvertical line is a number that measures how steeply the line is slanted (rise-over-run). If the line is the graph of the linear function f(x) = ax + b, this slope is given by the constant a. The slope measures the constant rate of change of f ( x ) {displaystyle f(x)} per unit change in x: whenever the input x is increased by one unit, the output changes by a units: f ( x + 1 ) = f ( x ) + a {displaystyle f(x{+}1)=f(x)+a} , and more generally f ( x + Δ x ) = f ( x ) + a Δ x {displaystyle f(x{+}Delta x)=f(x)+aDelta x} for any number Δ x {displaystyle Delta x} . If the slope is positive, a > 0 {displaystyle a>0} , then the function f ( x ) {displaystyle f(x)} is increasing; if a < 0 {displaystyle a<0} , then f ( x ) {displaystyle f(x)} is decreasing