Constant function
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 is a constant function because the value of is 4 regardless of the input value (see image).
![](http://up.wiki.x.io/wikipedia/commons/d/d3/Wiki_constant_function_175_200.png)
Basic properties
changeFormally, a constant function f(x):R→R has the form . Usually we write or just .
- 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 or just is the specific constant function where the output value is . 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 is a horizontal line in the plane that passes through the point .[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
changeIn 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: .
Example: is a constant function. The derivative of y is the identically zero function .
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:
Generalization
changeA function f : A → B is a constant function if f(a) = f(b) for every a and b in A.[8]
Examples
changeReal-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 : 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 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.
Generalized constant function. |
Constant function z(x,y)=2 |
Constant polar function ρ(φ)=2.5 |
Other properties
changeThere are other properties of constant functions.
Related pages
changeReferences
change- ↑ Tanton, James (2005). Encyclopedia of Mathematics. Facts on File, New York. p. 94. ISBN 0-8160-5124-0. (in English)
- ↑ C.Clapham, J.Nicholson (2009). "Oxford Concise Dictionary of Mathematics, Constant Function" (PDF). Addison-Wesley. p. 175. Retrieved 1 January 2014.
- ↑ Weisstein, Eric (1999). CRC Concise Encyclopedia of Mathematics. CRC Press, London. p. 313. ISBN 0-8493-9640-9. (in English)
- ↑ Dawkins, Paul (2007). "College Algebra". Lamar University. p. 224. Retrieved 1 January 2014.
- ↑ 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)
- ↑ Dawkins, Paul (2007). "Derivative Proofs". Lamar University. Retrieved 1 January 2014.
- ↑ "Zero Derivative implies Constant Function". Retrieved 1 January 2014.
- ↑ "Constant Function". Retrieved 1 January 2014.[permanent dead link]
Other websites
change- "Constant Function". From MathWorld--A Wolfram Web Resource. Retrieved 1 January 2014.