Whenever we want to discover the point of discontinuity of any function, we just have to equal the denominator to zero.

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## 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.

- \(\color{blue}{f(x)=\frac{x+2}{x^2-5x-6}}\)
- \(\color{blue}{f(x)=\frac{x-2}{x^2-2x-35}}\)
- \(\color{blue}{f(x)=\frac{x^2-6x+8}{x-5}}\)
- \(\color{blue}{f(x)=\frac{x+10}{x^2-10x+21}}\)

- \(\color{blue}{x=-1, x=6}\)
- \(\color{blue}{x=7, x=-5}\)
- \(\color{blue}{x=5}\)
- \(\color{blue}{x=3, x=7}\)

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