Examples of singularities in the following topics:
-
- 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:
-
- 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 $x=-1$.
-
- are continuous - there are no singularities or discontinuities.
-
- This is called a singular matrix.