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Concept
Constant function
Summary
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 regardless of the input value x (see image).
As a real-valued function of a real-valued argument, a constant function has the general form y(x) = c or just y = c.
Example: The function y(x) = 2 or just y = 2 is the specific constant function where the output value is c = 2. The domain of this function is the set of all real numbers R. The codomain of this function is just {2}. 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) = 2, y(−2.7) = 2, y(π) = 2, and so on. No matter what value of x is input, the output is "2".
Real-world example: A store where every item is sold for the price of 1 dollar.
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 where c is nonzero. 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: . The converse is also true. Namely, if y′(x) = 0 for all real numbers x, then y is a constant function.
Example: Given the constant function . The derivative of y is the identically zero function .
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