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.
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Whenever we want to discover the point of discontinuity of any function, we just have to set the denominator to zero.
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A step-by-step guide to the discontinuities of rational functions
In rational functions, points of discontinuity refer to fractions that are undefined 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.
- \(\color{blue}{f(x)=\frac{x+2}{x^2-5x-6}}\)
- \(\color{blue}{f(x)=\frac{x-2}{x^2-2x-35}}\)
- \(\color{blue}{f(x)=\frac{x^2-6x+8}{x-5}}\)
- \(\color{blue}{f(x)=\frac{x+10}{x^2-10x+21}}\)
- \(\color{blue}{x=-1, x=6}\)
- \(\color{blue}{x=7, x=-5}\)
- \(\color{blue}{x=5}\)
- \(\color{blue}{x=3, x=7}\)
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