constant function

(noun)

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

Related Terms

  • decreasing function
  • increasing function
  • composite function

Examples of constant function in the following topics:

  • Increasing, Decreasing, and Constant Functions

    • Functions can either be constant, increasing as xxx increases, or decreasing as xx 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)=cf(x)=cf(x)=c for all values of xxx and some constant ccc.  
    • 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
  • Introduction to Rational Functions

    • A rational function is one such that f(x)=P(x)Q(x)f(x) = \frac{P(x)}{Q(x)}f(x)=​Q(x)​​P(x)​​, where Q(x)≠0Q(x) \neq 0Q(x)≠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, xxx, 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)=1Q(x) = 1Q(x)=1.
    • A constant function such as f(x)=πf(x) = \pif(x)=π is a rational function since constants are polynomials.
  • Stretching and Shrinking

    • This is accomplished by multiplying either xxx or yyy by a constant, respectively.
    • Multiplying the entire function f(x)f(x)f(x) by a constant greater than one causes all the yyy values of an equation to increase.
    • where f(x)f(x)f(x) is some function and bbb is an arbitrary constant.  
    • Multiplying the independent variable xxx by a constant greater than one causes all the xxx values of an equation to increase.
    • where f(x)f(x)f(x) is some function and ccc 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)f(x)f(x) is some given function and bbb 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)f(x)f(x) would be the original function, and aaa is the constant being added or subtracted to cause a horizontal shift.  
  • Graphs of Exponential Functions, Base e

    • The function f(x)=exf(x) = e^xf(x)=e​x​​ is a basic exponential function with some very interesting properties.
    • The basic exponential function, sometimes referred to as the exponential function, is f(x)=exf(x)=e^{x}f(x)=e​x​​, where eee is the number (approximately 2.718281828) described previously.
    • y=exy=e^xy=e​x​​ is the only function with this property.
    • The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change (i.e., percentage increase or decrease) in the dependent variable.
    • Also, because the the growth rate of a population of bacteria in a petri dish is proportional to its size, the number of bacteria in the dish at a given time can be modeled by an exponential function such as y=Aekty=Ae^{kt}y=Ae​kt​​ where AAA is the number of bacteria present initially (at time t=0t=0t=0) and kkk is a constant called the growth constant.
  • What is a Quadratic Function?

    • where aaa, bbb, and ccc are constants and xxx is the independent variable.  
    • The constants bbband ccc can take any finite value, and aaa can take any finite value other than 000.
    • 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 xxx.  
    • With a linear function, each input has an individual, unique output (assuming the output is not a constant).  
  • What is a Linear Function?

    • A linear function is an algebraic equation in which each term is either a constant or the product of a constant and (the first power of) a single variable.
    • For example, a common equation, y=mx+by=mx+by=mx+b, (namely the slope-intercept form, which we will learn more about later) is a linear function because it meets both criteria with xxx and yyy as variables and mmm and bbb as constants.  
    • In the linear function graphs below, the constant, mmm, determines the slope or gradient of that line, and the constant term, bbb, determines the point at which the line crosses the yyy-axis, otherwise known as the yyy-intercept.
    • The blue line, y=12x−3y=\frac{1}{2}x-3y=​2​​1​​x−3 and the red line, y=−x+5y=-x+5y=−x+5 are both linear functions.  
    • Identify what makes a function linear and the characteristics of a linear function
  • Natural Logarithms

    • The natural logarithm is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.718281828.
    • Just as the exponential function with base eee arises naturally in many calculus contexts, the natural logarithm, which is the inverse function of the exponential with base eee also arises in naturally in many contexts.
    • It is used much more frequently in physics, chemistry, and higher mathematics than other logarithmic functions.
    • For example, the doubling time for a population which is growing exponentially is usually given as ln2k{\ln 2 \over k}​k​​ln2​​ where kkk is the growth rate, and the half-life of a radioactive substance is usually given as ln2λ{\ln 2 \over \lambda}​λ​​ln2​​ where λ\lambdaλ is the decay constant.
    • The natural logarithm function can be used to solve equations in which the variable is in an exponent.
  • Basics of Graphing Polynomial Functions

    • Another easy point to draw is the intersection with the yyy-axis, as this equals the function value in the point zero, which equals the constant term of the polynomial.
    • Conversely, we can easily read the constant term of the polynomial by looking at its intersection with the yyy-axis if its graph is given (and indeed, we can readily read any function value if the graph is given).
    • The graph of a non-constant (univariate) polynomial always tends to infinity when the variable increases indefinitely (in absolute value).
    • Its constant term is between -1 and 0.
    • Its constant term is between 3 and 4.
  • Completing the Square

    • The value of kkk is meant to adjust the function to compensate for the difference between the expanded form of a(x−h)2a(x-h)^2a(x−h)​2​​ and the general quadratic function ax2+bx+cax^2+bx+cax​2​​+bx+c.  
    • However, it is possible to write the original quadratic as the sum of this square and a constant:
    • Thus, the constant hhh takes the value −5-5−5 and the constant kkk takes the value −3-3−3.
    • Solve for the zeros of a quadratic function by completing the square
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