Root of a Function

A Root of a Function is a x-coordinate for which $f(x)=0$ is satisfied.

• AKA: Zero of a Function.
• Context:
  • It can range from being a Real-valued Root to being a Complex-valued Root.
• Example(s):
  • a Polynomial Root,
  • …
• Counter-Example(s):
  • Function Maximum,
  • Function Minimum.
• See: Root-Finding Algorithm, Domain of a Function, Equation Solution, Polynomial Function, Fundamental Theorem of Algebra, Degree of a Polynomial.

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References

2021

• (MathWorld, 2021) ⇒ https://mathworld.wolfram.com/Root.html Retrieved:2021-09-05.
  • QUOTE: The roots (sometimes also called “zeros") of an equation
    $f(x)=0$
    are the values of $x$ for which the equation is satisfied.

2015

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/Zero_of_a_function Retrieved:2015-11-18.
  • In mathematics, a zero, also sometimes called a root, of a real-, complex- or generally vector-valued function f is a member x of the domain of f such that f(x) vanishes at x ; that is, x is a solution of the equation f(x) = 0. In other words, a "zero" of a function is an input value that produces an output of zero (0).

    A root of a polynomial is a zero of the associated polynomial function.

    The fundamental theorem of algebra shows that any non-zero polynomial has a number of roots at most equal to its degree and that the number of roots and the degree are equal when one considers the complex roots (or more generally the roots in an algebraically closed extension) counted with their multiplicities. For example, the polynomial f of degree two, defined by :

    [math]\displaystyle{ f(x)=x^2-5x+6 }[/math] has the two roots 2 and 3, since :

    [math]\displaystyle{ f(2) = 2^2 - 5 \cdot 2 + 6 = 0 \quad \textstyle{\rm {and} }\quad f(3) = 3^2 - 5 \cdot 3 + 6 = 0. }[/math]

    If the function maps real numbers to real numbers, its zeroes are the x-coordinates of the points where its graph meets the x-axis. An alternative name for such a point (x,0) in this context is an x-intercept.

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