singularities

(noun)

The $x$-values at which a rational function is not defined, for which the denominator $Q(x)$ is zero.

Related Terms

  • Example
  • domain
  • rational function
  • vertical asymptote
  • denominator

Examples of singularities in the following topics:

  • Introduction to Rational Functions

    • Domain restrictions can be calculated by finding singularities, which are the $x$-values for which the denominator $Q(x)$ is zero.
    • Factorizing the numerator and denominator of rational function helps to identify singularities of algebraic rational functions.
    • Singularity occurs when the denominator of a rational function equals $0$, whether or not the linear factor in the denominator cancels out with a linear factor in the numerator.
    • We can factor the denominator to find the singularities of the function:
  • Asymptotes

    • In other words, vertical asymptotes occur at singularities, or points at which the rational function is not defined.
    • Vertical asymptotes only occur at singularities when the associated linear factor in the denominator remains after cancellation.
    • We can identify from the linear factors in the denominator that two singularities exist, at $x=1$ and $x = -1$.
    • Notice that, based on the linear factors in the denominator, singularities exists at $x=1$ and $x=-1$.
  • The Intermediate Value Theorem

    • are continuous - there are no singularities or discontinuities.
  • The Inverse of a Matrix

    • This is called a singular matrix.
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