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Constant function y=4

In mathematics, a constant function is a function whose output value is the same for every input value.[1][2][3] For example, the function [math]y(x) = 4[/math] is a constant function because the value of  [math]y(x)[/math]  is 4 regardless of the input value [math]x[/math] (see image).

Basic properties

Formally, a constant function f(x):RR has the form  [math]f(x)=c[/math]. Usually we write [math]y(x)=c[/math]  or just  [math]y=c[/math].

  • The function y=c has 2 variables x and у and 1 constant c. (In this form of the function, we do not see x, but it is there.)
    • The constant c is a real number. Before working with a linear function, we replace c with an actual number.
    • The domain or input of y=c is R. So any real number x can be input. However, the output is always the value c.
    • The range of y=c is also R. However, because the output is always the value of c, the codomain is just c.

Example: The function  [math]y(x)=4[/math]  or just  [math]y=4[/math]  is the specific constant function where the output value is  [math]c=4[/math]. The domain is all real numbers ℝ. The codomain is just {4}. 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 [math]y=c[/math] is a horizontal line in the plane that passes through the point [math](0,c)[/math].[4]
  • If c≠0, the constant function y=c is a polynomial in one variable x of degree zero.
    • The y-intercept of this function is the point (0,c).
    • This function has no x-intercept. That is, it has no root or zero. It never crosses the x-axis.
  • If c=0, then we have y=0. This is the zero polynomial or the identically zero function. Every real number x is a root. The graph of y=0 is the x-axis in the plane.[5]
  • A constant function is an even function so the y-axis is an axis of symmetry for every constant function.

Derivative of a constant function

In the context where it is defined, the derivative of a function measures the rate of change of function (output) values with respect to change in input values. A constant function does not change, so its derivative is 0.[6] This is often written:  [math](c)'=0[/math] .

Example:  [math]y(x)=-\sqrt{2}[/math]  is a constant function. The derivative of y is the identically zero function   [math]y'(x)=(-\sqrt{2})'=0[/math]  .

The converse (opposite) is also true. That is, if the derivative of a function is zero everywhere, then the function is a constant function.[7]

Mathematically we write these two statements:

[math]y(x)=c \,\,\, \Leftrightarrow \,\,\, y'(x)=0 \,, \,\,\forall x \in \mathbb{R}[/math]

Generalization

A function f : AB is a constant function if f(a) = f(b) for every a and b in A.[8]

Examples

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

Example: Let f : AB where A={X,Y,Z,W} and B={1,2,3} and f(a)=3 for every aA. 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 3-dimensional space that passes through the point (0,0,2).

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.

Constant function gen2.svg
Generalized constant function.
Constant function plane.png
Constant function z(x,y)=2
Constant function polar.png
Constant polar function ρ(φ)=2.5

Other properties

There are other properties of constant functions. See Constant function on English Wikipedia

Related pages

References

  1. Tanton, James (2005). Encyclopedia of Mathematics. Facts on File, New York. p. 94. ISBN 0-8160-5124-0 .  (in English)
  2. C.Clapham, J.Nicholson (2009). "Oxford Concise Dictionary of Mathematics, Constant Function" (in English). Addison-Wesley. p. 175. http://web.cortland.edu/matresearch/OxfordDictionaryMathematics.pdf. Retrieved January 2014. 
  3. Weisstein, Eric (1999). CRC Concise Encyclopedia of Mathematics. CRC Press, London. p. 313. ISBN 0-8493-9640-9 .  (in English)
  4. Dawkins, Paul (2007). "College Algebra" (in English). Lamar University. p. 224. http://tutorial.math.lamar.edu/Classes/Alg/Alg.aspx. Retrieved January 2014. 
  5. Carter, John A.; Cuevas, Gilbert J.; Holliday, Berchie; Marks, Daniel; McClure, Melissa S.publisher=Glencoe/McGraw-Hill School Pub Co (2005). Advanced Mathematical Concepts - Pre-calculus with Applications, Student Edition 1. p. 22. ISBN 978-0078682278 .  (in English)
  6. Dawkins, Paul (2007). "Derivative Proofs" (in English). Lamar University. http://tutorial.math.lamar.edu/Classes/CalcI/DerivativeProofs.aspx. Retrieved January 2014. 
  7. "Zero Derivative implies Constant Function" (in English). http://www.proofwiki.org/wiki/Zero_Derivative_implies_Constant_Function. Retrieved January 2014. 
  8. "Constant Function" (in English). http://planetmath.org/ConstantFunction.html. Retrieved January 2014. 

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