Constant function

Constant function y=4

In mathematics, a constant function is a function whose (output) value is the same for every input value. For example, the function $y(x)=4$ is a constant function because the value of  $y(x)$  is 4 for every input value $x$ (see image).

Basic properties

As a real-valued function of a real-valued argument, a constant function has the form  $y(x)=c$  or just  $y=c$  .

Example: The function  $y(x)=4$  or just  $y=4$  is the specific constant function where the output value is  $c=4$. The domain of this function is the set of all real numbers ℝ. The codomain of this function is just {4}. The independent variable x does not appear on the right side of the function expression and so its value is vacuously substituted. Namely y(0)=4, y(−2.7)=4, y(π)=4,.... No matter what value of x is input, the output is "4".

The graph of the constant function $y=c$ is a horizontal line in the plane that passes through the point $(0,c)$.

In the context of a polynomial in one variable x, the non-zero constant function is a polynomial of degree 0 and its general form is $f(x)=c\,,\,\,c\neq 0$ . This function has no intersection point with the x-axis, that is, it has no root (zero). On the other hand, the polynomial  $f(x)=0$  is the identically zero function. It is the (trivial) constant function and every x is a root. Its graph is the x-axis in the plane.

A constant function is an even function, i.e. the graph of a constant function is symmetric with respect to the y-axis.

In the context where it is defined, the derivative of a function is a measure of the rate of change of function values with respect to change in input values. Because a constant function does not change, its derivative is 0. This is often written:  $(c)'=0$ . The converse (opposite) is also true. Namely, if y'(x)=0 for all real numbers x, then y(x) is a constant function.

Example: Given the constant function   $y(x)=-{\sqrt {2}}$  . The derivative of y is the identically zero function   $y'(x)=(-{\sqrt {2}})'=0$  .

Generalization

A function f : A → B is a constant function if f(a) = f(b) for every a and b in A.

Examples

Real-world example: A store where every item is sold for the price of 1 euro. The domain of this function is the set of items in the store. The codomain is the set {1 euro}.

Example: Let f : A → B where A={X,Y,Z,W} and B={1,2,3} and f(a)=3 for every a∈A. Then f is a constant function.

Example: z(x,y)=2 is the constant function from A=ℝ² to B=ℝ where every point (x,y)∈ℝ² is mapped to the value z=2. The graph of this constant function is the horizontal plane (parallel to the x0y plane) in 3D space that passes through the point (0,0,2). The function z(x,y)=0 is the identically zero function whose graph is the x0y plane in 3D space.

Example: The polar function ρ(φ)=2.5 is the constant function that maps every angle φ to the radius ρ=2.5. The graph of this function is the circle of radius 2.5 in the plane.

Generalized constant function.

Constant function z(x,y)=2

Constant polar function ρ(φ)=2.5

Other properties

There are other properties associated with constant functions. See Constant function on Wikipedia

See also

• Function (mathematics)
• Polynomial

Other websites

• Weisstein, Eric W. "Constant Function". From MathWorld--A Wolfram Web Resource. Retrieved January 2014. {{cite web}}: Check date values in: |accessdate= (help); Cite has empty unknown parameter: |1= (help)