Domain Of A Constant Function

Type of mathematical function

Constant function y=4

In mathematics, a abiding function is a role whose (output) value is the aforementioned for every input value.^[ane] ^[2] ^[3] For example, the function $y (ten) = 4$ is a constant function because the value of $y (x)$ is iv regardless of the input value $x$ (see epitome).

Bones properties [edit]

Equally a existent-valued part of a real-valued argument, a constant office has the full general grade $y (10) = c$ or just $y = c$ . ^[4]

Example: The office

y (x) = 2

or just

y = 2

is the specific constant function where the output value is

c = ii

. The domain of this function is the ready of all real numbers R. The codomain of this office is just {2}. The contained variable x does not appear on the correct side of the office expression and and then its value is "vacuously substituted". Namely

y (0) = ii

y (-two.7) = two

y (π) = 2

, and then on. No matter what value of x is input, the output is "2".

Real-world example: A shop where every item is sold for the price of 1 dollar.

The graph of the constant role $y = c$ is a horizontal line in the plane that passes through the point $(0, c)$ . ^[5]

In the context of a polynomial in ane variable ten, the not-goose egg constant function is a polynomial of degree 0 and its general course is $f (10) = c$ where $c$ is nonzero. This function has no intersection indicate with the x-centrality, that is, it has no root (zero). On the other hand, the polynomial $f (10) = 0$ is the identically cipher function. It is the (trivial) constant office and every x is a root. Its graph is the x-axis in the plane.^[6]

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

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 role does non change, its derivative is 0.^[7] This is often written: $(ten\mapsto c)'=0$ . The antipodal is also truthful. Namely, if $y'(x) = 0$ for all real numbers ten, then y is a constant office.^[eight]

Example: Given the constant function

y(x)=-{\sqrt {two}}

. The derivative of y is the identically zero function

y'(x)=\left(x\mapsto -{\sqrt {two}}\correct)'=0

Other backdrop [edit]

For functions betwixt preordered sets, constant functions are both guild-preserving and society-reversing; conversely, if f is both social club-preserving and order-reversing, and if the domain of f is a lattice, and then f must be constant.

Every abiding function whose domain and codomain are the aforementioned set X is a left aught of the full transformation monoid on 10, which implies that it is also idempotent.
It has cypher gradient/gradient.
Every constant function betwixt topological spaces is continuous.
A constant role factors through the ane-point set, the terminal object in the category of sets. This observation is instrumental for F. William Lawvere'south axiomatization of set theory, the Unproblematic Theory of the Category of Sets (ETCS).^[9]
For whatsoever non-empty Y, every prepare 10 is isomorphic to the set of constant functions in $Y\to 10$ $Y\to X$ . For any Y and each chemical element x in X, at that place is a unique part ${\tilde {x}}:Y\to X$ ${\tilde {x}}:Y\to X$ such that ${\tilde {x}}(y)=x$ ${\tilde {x}}(y)=x$ for all $y\in Y$ $y\in Y$ . Conversely, if a function $f:Y\to X$ $f:Y\to X$ satisfies $f(y)=f\left(y'\right)$ $f(y)=f\left(y'\right)$ for all $y,y'\in Y$ $y,y'\in Y$ , $f$ $f$ is by definition a constant role.
- Equally a corollary, the 1-indicate set is a generator in the category of sets.
- Every set $X$ is canonically isomorphic to the role set $X^{1}$ , or hom set $\operatorname {hom} (1,X)$ in the category of sets, where one is the one-point set. Because of this, and the adjunction betwixt Cartesian products and hom in the category of sets (and then there is a canonical isomorphism between functions of ii variables and functions of ane variable valued in functions of some other (single) variable, $\operatorname {hom} (X\times Y,Z)\cong \operatorname {hom} (X(\operatorname {hom} (Y,Z))$ ) the category of sets is a airtight monoidal category with the Cartesian production of sets equally tensor product and the one-betoken ready as tensor unit of measurement. In the isomorphisms $\lambda :1\times X\cong X\cong X\times 1:\rho$ natural in X, the left and right unitors are the projections $p_{1}$ and $p_{two}$ the ordered pairs $(*,x)$ and $(x,*)$ respectively to the element $x$ , where $*$ is the unique point in the 1-point set.

A function on a continued set is locally constant if and just if it is constant.

References [edit]

^ Tanton, James (2005). Encyclopedia of Mathematics. Facts on File, New York. p. 94. ISBN0-8160-5124-0.
^ C.Clapham, J.Nicholson (2009). "Oxford Concise Dictionary of Mathematics, Constant Role" (PDF). Addison-Wesley. p. 175. Retrieved January 12, 2014.
^ Weisstein, Eric (1999). CRC Concise Encyclopedia of Mathematics. CRC Press, London. p. 313. ISBN0-8493-9640-ix.
^ Weisstein, Eric W. "Constant Function". mathworld.wolfram.com . Retrieved 2020-07-27 .
^ Dawkins, Paul (2007). "College Algebra". Lamar University. p. 224. Retrieved January 12, 2014.
^ Carter, John A.; Cuevas, Gilbert J.; Holliday, Berchie; Marks, Daniel; McClure, Melissa Southward. (2005). "1". Advanced Mathematical Concepts - Pre-calculus with Applications, Student Edition (1 ed.). Glencoe/McGraw-Hill Schoolhouse Pub Co. p. 22. ISBN978-0078682278.
^ Dawkins, Paul (2007). "Derivative Proofs". Lamar Academy. Retrieved January 12, 2014.
^ "Zero Derivative implies Constant Function". Retrieved January 12, 2014.
^ Leinster, Tom (27 Jun 2011). "An informal introduction to topos theory". arXiv:1012.5647 [math.CT].