How to Find Discontinuities of Rational Functions?

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

Related Topics

• How to Find the End Behavior of Rational Functions
• How to Graph Rational Functions

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.

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

1. (color{blue}{x=-1, x=6})
2. (color{blue}{x=7, x=-5})
3. (color{blue}{x=5})
4. (color{blue}{x=3, x=7})