Question

How do you find discontinuity algebraically?

Answer 1

Hint: In this problem, we have to analyse how to find discontinuity algebraically. We can start with an example. We can take an equation with both numerator and the denominator, we can factorize it, to get factors in both numerator and the denominator. We can then find the common factors in the numerator and denominator to set zero for it and we can solve for x which is the point of discontinues, where discontinuity occurs.

Complete step by step solution:
We should know about discontinuous functions to find the discontinuity algebraically.
A discontinuous function is a function that is not a continuous curve that has points that are isolated from each other. If f(x) is not continuous at x = a, then f(x) is said to be discontinued at this point.
We should also know that a point of discontinuity occurs when a number given zero for both numerator and the denominator.
We can take an example.
\[f\left( x \right)=\dfrac{{{x}^{4}}+4x-5}{{{x}^{2}}+7x+10}\]
We can now factorize it, we get
\[f\left( x \right)=\dfrac{\left( x+5 \right)\left( x-1 \right)}{\left( x+5 \right)\left( x+2 \right)}=\dfrac{x-1}{x+2}\]
We can see the common factor in the both numerator and the denominator. We also know that discontinuity occurs when a number is given zero for both the numerator and the denominator.
Since, x = -5 is a zero for both numerator and the denominator for the above step, there is a point of discontinuity there.
We can find the y value by substituting the x value in the simplified form, we get
\[\Rightarrow \dfrac{-5-2}{-5+1}=2\]
Therefore, \[\left( -5,2 \right)\] is the point of discontinuity.

Note:
We should always remember that the point of discontinuity occurs when a number is both a zero of the numerator and the denominator when we have a common factor in both numerator and the denominator. We can then substitute the x value at discontinuity in the simplified form to get y value, therefore, we can get a point of discontinuity.