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.

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.

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. $$\color{blue}{x=3, x=7}$$

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