How Do You Find Points Of Discontinuity In Rational Functions? - Vedantu

CoursesCourses for KidsFree study materialOffline CentresMoreStore

Talk to our experts

1800-120-456-456

Sign In

• Question Answer

• Class 12

• Maths

• How do you find points of disc...

AnswerQuestion Answers for Class 12Class 12 BiologyClass 12 ChemistryClass 12 EnglishClass 12 MathsClass 12 PhysicsClass 12 Social ScienceClass 12 Business StudiesClass 12 EconomicsQuestion Answers for Class 11Class 11 EconomicsClass 11 Computer ScienceClass 11 BiologyClass 11 ChemistryClass 11 EnglishClass 11 MathsClass 11 PhysicsClass 11 Social ScienceClass 11 AccountancyClass 11 Business StudiesQuestion Answers for Class 10Class 10 ScienceClass 10 EnglishClass 10 MathsClass 10 Social ScienceClass 10 General KnowledgeQuestion Answers for Class 9Class 9 General KnowledgeClass 9 ScienceClass 9 EnglishClass 9 MathsClass 9 Social ScienceQuestion Answers for Class 8Class 8 ScienceClass 8 EnglishClass 8 MathsClass 8 Social ScienceQuestion Answers for Class 7Class 7 ScienceClass 7 EnglishClass 7 MathsClass 7 Social ScienceQuestion Answers for Class 6Class 6 ScienceClass 6 EnglishClass 6 MathsClass 6 Social ScienceQuestion Answers for Class 5Class 5 ScienceClass 5 EnglishClass 5 MathsClass 5 Social ScienceQuestion Answers for Class 4Class 4 ScienceClass 4 EnglishClass 4 MathsHow do you find points of discontinuity in rational functions?AnswerVerified565.2k+ viewsHint: We take an example to illustrate how to find the points of discontinuity. In order to find the point of discontinuity of any function, we simply equate the denominator with zero.Complete step-by-step solution:Points of discontinuity in rational functions refer to those fractions which are undefined or which have their denominators as zero. Once a fraction has its denominator is zero, it becomes undefined and then it appears as a hole or a break in the graph. Let us take a rational function as an example to illustrate this: Let us take $f\left( x \right) = \dfrac{{{x^2} + 2x - 8}}{{{x^2} + 5x + 4}}$Now we factories both the numerator and denominator, to do this we first split the middle terms:\[ \Rightarrow f\left( x \right) = \dfrac{{{x^2} + 4x - 2x - 8}}{{{x^2} + 4x + x + 4}}\]Now we group the common factors:$ \Rightarrow f\left( x \right) = \dfrac{{x\left( {x + 4} \right) - 2\left( {x + 4} \right)}}{{x\left( {x + 4} \right) + 1\left( {x + 4} \right)}}$Thus we successfully factories the numerator and denominator to get our factors as:$ \Rightarrow f\left( x \right) = \dfrac{{\left( {x + 4} \right)\left( {x - 2} \right)}}{{\left( {x + 4} \right)\left( {x + 1} \right)}}$ In the above expression, we find that $\left( {x + 4} \right)$ can be canceled out from both the denominator and the numerator, thus it becomes a removable discontinuity while $\left( {x + 1} \right)$ cannot be canceled out or factored any further hence it becomes non-removable discontinuity. Now, this rational function will only be discontinuous when the denominator is equal to zero. To find the discontinuities, we equate the denominator with zero. Thus, we get:$x + 4 = 0$ and $x + 1 = 0$ $ \Rightarrow x = - 4$ and $x = - 1$ , thus the given rational functions will be discontinuous at these following points. Note: Let us take another rational function as an example.Let $f\left( x \right) = \dfrac{{x + 1}}{{{x^2} - 5x - 6}}$ , now to prove discontinuity again we need to equate the denominator with zeroThus, we find the factors of the denominator$ \Rightarrow f\left( x \right) = \dfrac{{x + 1}}{{{x^2} - 6x + x - 6}}$ We do grouping to find the common factors:$ \Rightarrow f\left( x \right) = \dfrac{{x + 1}}{{\left( {x + 1} \right)\left( {x - 6} \right)}}$Thus, if we equate the factors in the denominator, we find that the function is discontinuous at $x = - 1$ and $x = 6$ Recently Updated PagesBasicity of sulphurous acid and sulphuric acid areMaster Class 12 English: Engaging Questions & Answers for SuccessMaster Class 12 Social Science: Engaging Questions & Answers for SuccessMaster Class 12 Maths: Engaging Questions & Answers for SuccessMaster Class 12 Economics: Engaging Questions & Answers for SuccessMaster Class 12 Physics: Engaging Questions & Answers for SuccessBasicity of sulphurous acid and sulphuric acid areMaster Class 12 English: Engaging Questions & Answers for SuccessMaster Class 12 Social Science: Engaging Questions & Answers for SuccessMaster Class 12 Maths: Engaging Questions & Answers for SuccessMaster Class 12 Economics: Engaging Questions & Answers for SuccessMaster Class 12 Physics: Engaging Questions & Answers for Success

• 1
• 2

Trending doubtsDraw a labelled sketch of the human eye class 12 physics CBSEWhich are the Top 10 Largest Countries of the World?Draw ray diagrams each showing i myopic eye and ii class 12 physics CBSEGiving reasons state the signs positive or negative class 12 physics CBSEExplain esterification reaction with the help of a class 12 chemistry CBSEWhat is defined as a solenoid Depict a diagram with class 12 physics CBSEDraw a labelled sketch of the human eye class 12 physics CBSEWhich are the Top 10 Largest Countries of the World?Draw ray diagrams each showing i myopic eye and ii class 12 physics CBSEGiving reasons state the signs positive or negative class 12 physics CBSEExplain esterification reaction with the help of a class 12 chemistry CBSEWhat is defined as a solenoid Depict a diagram with class 12 physics CBSE

• 1
• 2