constant function

(noun)

A function whose value is the same for all the elements of its domain.

Related Terms

Examples of constant function in the following topics:

  • Increasing, Decreasing, and Constant Functions

    • Functions can either be constant, increasing as $x$ increases, or decreasing as $x $ increases.
    • In mathematics, a constant function is a function whose values do not vary, regardless of the input into the function.  
    • A function is a constant function if $f(x)=c$ for all values of $x$ and some constant $c$.  
    • Example 1:  Identify the intervals where the function is increasing, decreasing, or constant.
    • Identify whether a function is increasing, decreasing, constant, or none of these

  • Derivatives of Exponential Functions

    • The derivative of the exponential function is equal to the value of the function.
    • Functions of the form $ce^x$ for constant $c$ are the only functions with this property.
    • The rate of increase of the function at $x$ is equal to the value of the function at $x$.
    • If a variable's growth or decay rate is proportional to its size—as is the case in unlimited population growth, continuously compounded interest, or radioactive decay—then the variable can be written as a constant times an exponential function of time.
    • Explicitly for any real constant $k$, a function $f: R→R$ satisfies $f′ = kf $ if and only if $f(x) = ce^{kx}$ for some constant $c$.

  • Introduction to Rational Functions

    • A rational function is one such that $f(x) = \frac{P(x)}{Q(x)}$, where $Q(x) \neq 0$; the domain of a rational function can be calculated.
    • A rational function is any function which can be written as the ratio of two polynomial functions.
    • Any function of one variable, $x$, is called a rational function if, and only if, it can be written in the form:
    • Note that every polynomial function is a rational function with $Q(x) = 1$.
    • A constant function such as $f(x) = \pi$ is a rational function since constants are polynomials.

  • Differentiation Rules

    • If $f(x)$ is a constant, then $f'(x) = 0$, since the rate of change of a constant is always zero.
    • By extension, this means that the derivative of a constant times a function is the constant times the derivative of the function.
    • for all functions $f$ and $g$ at all inputs where $g \neq 0$.
    • The known derivatives of the elementary functions $x^2$, $x^4$, $\ln(x)$, and $e^x$, as well as that of the constant 7, were also used.

  • Stretching and Shrinking

    • This is accomplished by multiplying either $x$ or $y$ by a constant, respectively.
    • Multiplying the entire function $f(x)$ by a constant greater than one causes all the $y$ values of an equation to increase.
    • where $f(x)$ is some function and $b$ is an arbitrary constant.  
    • Multiplying the independent variable $x$ by a constant greater than one causes all the $x$ values of an equation to increase.
    • where $f(x)$ is some function and $c$ is an arbitrary constant.  

  • Translations

    • A translation moves every point in a function a constant distance in a specified direction.
    • To translate a function vertically is to shift the function up or down.
    • where $f(x)$ is some given function and $b$ is the constant that we are adding to cause a translation.
    • To translate a function horizontally is the shift the function left or right.
    • Where $f(x)$ would be the original function, and $a$ is the constant being added or subtracted to cause a horizontal shift.  

  • Expressing Functions as Power Functions

    • A power function is a function of the form $f(x) = cx^r$ where $c$ and $r$ are constant real numbers.
    • A power function is a function of the form $f(x) = cx^r$ where $c$ and $r$ are constant real numbers.
    • Polynomials are made of power functions.
    • Functions of the form $f(x) = x^3$, $f(x) = x^{1.2}$, $f(x) = x^{-4}$ are all power functions.
    • Describe the relationship between the power functions and infinitely differentiable functions

  • What is a Quadratic Function?

    • where $a$, $b$, and $c$ are constants and $x$ is the independent variable.  
    • The constants $b$and $c$ can take any finite value, and $a$ can take any finite value other than $0$.
    • A quadratic equation is a specific case of a quadratic function, with the function set equal to zero:
    • When all constants are known, a quadratic equation can be solved as to find a solution of $x$.  
    • With a linear function, each input has an individual, unique output (assuming the output is not a constant).  

  • Solving Differential Equations

    • Solving the differential equation means solving for the function $f(x)$.
    • Solving this equation shows that $f(x)$ is equal to the negative of its derivative; therefore, the function $f(x)$ must be $e^{-x}$, as the derivative of this function equals the negative of the original function.
    • Therefore, the general solution is $f(x) = Ce^{-x}$, where $C$ stands for an arbitrary constant.
    • You can see that the differential equation still holds true with this constant.
    • For a specific solution, replace the constants in the general solution with actual numeric values.

  • Expressing the Equilibrium Constant of a Gas in Terms of Pressure

    • Up to this point, we have been discussing equilibrium constants in terms of concentration.
    • Our equilibrium constant in terms of partial pressures, designated KP, is given as:
    • Note that in order for K to be constant, temperature must be constant as well.
    • Therefore, the term RT is a constant in the above expression.
    • The internal pressure of the gaseous propane is a function of temperature.