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
- How to Classify Numbers?
- Top 10 3rd Grade NYSE Math Practice Questions
- The Ultimate 7th Grade WVGSA Math Course (+FREE Worksheets)
- How to Factor Numbers? (+FREE Worksheet!)
- What is a Good SAT Score?
- What to Look for in Graphing Calculators?
- Exploring the Fundamentals: Properties of Equality and Congruence in Geometry
- The 8 Best Webcams for Online Teachers
- Top 10 Tips You MUST Know to Retake the SSAT Math
- SSAT Middle-Level Math Worksheets: FREE & Printable
What people say about "How to Find Discontinuities of Rational Functions? - Effortless Math: We Help Students Learn to LOVE Mathematics"?
No one replied yet.