How to Find Discontinuities of Rational Functions?
Discontinuities of rational functions occur when the denominator is \(0\). Read this post to know more about finding discontinuities of rational functions.

Whenever we want to discover the point of discontinuity of any function, we just have to equal the denominator to zero.
Related Topics
A step-by-step guide to the discontinuities of rational functions
In rational functions, points of discontinuity refer to fractions that are undefinable or have zero denominators. When the denominator of a fraction is \(0\), it becomes undefined and appears as a whole or a break in the graph.
To find discontinuities of rational functions, follow these steps:
- Obtain a function’s equation. Note that if the numerator and denominator expressions have any similar factors, they should be wiped out.
- Rewrite the denominator expression as a zero-valued equation.
- Solve the equation for the denominator.
The Discontinuities of Rational Functions – Example 1:
Find the discontinuities of \(f(x)=\frac{x-1}{x^2-x-6}\).
First, setting the denominator equal to zero: \(x^2-x-6=0\).
Then factoring it out: \(x^2-x-6=0\) ⇒ \((x+2)(x-3)=0\)
\(x+2=0 ⇒ x=-2\)
\(x-3=0 ⇒x=3\)
Now, \(f\) is discontinuous at \(x=-2\) and \(x=3\).
The Discontinuities of Rational Functions – Example 2:
Find the discontinuities of \(f(x)=\frac{1}{x^2-4}\).
First, setting the denominator equal to zero: \(x^2-4=0\).
Then factoring it out: \(x^2-4=0\) ⇒ \((x+2)(x-2)\).
\(x+2=0\) ⇒ \(x=-2\)
\(x-2=0\) ⇒ \(x=2\)
Now, \(f\) is discontinuous at \(x=-2\) and \(x=2\).
Exercises for the Discontinuities of Rational Functions
Find the discontinuities of rational functions.
- \(\color{blue}{f(x)=\frac{x+2}{x^2-5x-6}}\)
- \(\color{blue}{f(x)=\frac{x-2}{x^2-2x-35}}\)
- \(\color{blue}{f(x)=\frac{x^2-6x+8}{x-5}}\)
- \(\color{blue}{f(x)=\frac{x+10}{x^2-10x+21}}\)

- \(\color{blue}{x=-1, x=6}\)
- \(\color{blue}{x=7, x=-5}\)
- \(\color{blue}{x=5}\)
- \(\color{blue}{x=3, x=7}\)
Related to This Article
More math articles
- CLEP College Mathematics Worksheets: FREE & Printable
- 10 Most Common 7th Grade STAAR Math Questions
- 4th Grade STAAR Math Practice Test Questions
- 8th Grade ACT Aspire Math Worksheets: FREE & Printable
- FREE 4th Grade PARCC Math Practice Test
- A Comprehensive Collection of Free ATI TEAS 7 Math Practice Tests
- Top 10 4th Grade ACT Aspire Math Practice Questions
- The Ultimate 7th Grade Common Core Math Course (+FREE Worksheets)
- How is the ATI TEAS 7 Test Scored?
- The Math Expedition: How to Use Theoretical Probability to Predict the Unpredictable
What people say about "How to Find Discontinuities of Rational Functions? - Effortless Math: We Help Students Learn to LOVE Mathematics"?
No one replied yet.