# Undefined

Fractions can be undefined. An undefined fraction is a fraction that doesn't make sense.  Simply: an undefined fraction is a fraction with a zero denominator.

If a have a pizza and cut it into 6 pieces, I can give you 1 piece ($\dfrac{1}{6}$) or 4 pieces ($\dfrac{4}{6}$).

I could also give you no pizza.  We would write that fraction as $\dfrac{0}{6}$, or zero sixths.  There are six pieces, I gave you 0. A fraction with a zero in the numerator is equal to zero.  It is a real number.

While $\dfrac{0}{6}$ equals zero, what does $\dfrac{6}{0}$ equal?

It's undefined.  Why?

How can you cut a pizza into "zeroth" pieces?  How can you give someone 6 "zeroths" of a pizza?  You can't.  A fraction with a zero in the denominator is not a real number: it's undefined.

So, an undefined fraction is just a fraction with a zero in the denominator.  Any time you are asked what value makes a fraction undefined, it's the value that makes the denominator zero.

Example:

What value of $x$ makes $\dfrac{3x-1}{3x+4}$ undefined?

In this case, you just want to find what makes the denominator ${3x+4}$ equal to zero. You can ignore the numerator entirely, because all you need to make a fraction undefined is a denominator that is equal to zero.

\eqalign{3x+4&=0\\-4 &\quad-4\\3x&=-4\\x&=\dfrac{-4}{3}}

Let's plug that value of $x$ into the fraction and see if it works:

\eqalign{\dfrac{3x-1}{3x+4}=\dfrac{3(\dfrac{-4}{3})-1}{3(\dfrac{-4}{3})+4}=\dfrac{\dfrac{-12}{3}-1}{\dfrac{-12}{3}+4}=\dfrac{-4-1}{-4+4}=\dfrac{-5}{0}}

The denominator is zero.  The fraction is undefined.  The value of $x$ that make the fraction undefined is $\dfrac{-4}{3}$.

• ## Undefined

For the following problems, for what value(s) of x is each function undefined?

1. $f(x)=\dfrac{1}{8x}$

2. $f(x)=\dfrac{3x}{x-16}$

3. $f(x)=\dfrac{x+12}{x^2+7x+6}$

4. $f(x)=\dfrac{1}{\dfrac{x}{7}-3}$

5. $f(x)=\dfrac{82}{12x^2+31x+7}$

6. $f(x)=\dfrac{2x}{x^3-27}$