Defined & undefined

You’ll see the terms defined, undefined, and not defined in many GRE problems. We’ll briefly define them here, then focus on the situations where they matter most on the test.

Definitions

Undefined

An expression or quantity is undefined (or not defined) when it does not have a valid value under the given rules or conditions. A function value or sequence term can also be not defined if the rule does not assign a value to that input or position.

Defined

An expression or quantity is defined when it has a valid value under the given rules or conditions.

Undefined

There are four common situations on the GRE in which a quantity is not defined:

1. The result of dividing any number by $0$ is undefined.

$$x/0=\text{undefined}$$

2. On the GRE, the square root of any negative number is not defined.

$$\begin{array}{rl}\sqrt{x}& =\text{undefined}\\ \text{when:}& \\ x& <0\end{array}$$

3. Zero to the zero power is undefined.

$$0^0=\text{undefined}$$

4. A vertical line (one that goes straight up, like $x=5$) has an undefined slope. The slope is “rise/run.” On a vertical line, the $x$-value never changes, so the “run” (the change in $x$) is $0$. Since you can’t divide by $0$, the slope is undefined.

Knowing when expressions become undefined is especially important in problems where a variable could take multiple values. Here’s an example:

$x^2+7x=0$
$y/x=z$

$z$ is defined
$z<x$

Quantity A: $x$
Quantity B: $y$

Start by factoring the first equation:

$$\begin{array}{rl}{x}^2+7x& =0\\ x(x+7)& =0\end{array}$$

So $x=0$ or $x=-7$.

Now use the second equation: $y/x=z$. Here, $x$ is in the denominator. If $x=0$, then $y/x$ would involve division by $0$, which is undefined. But the problem tells you that $z$ is defined, so $x$ cannot be $0$.

That means the only possible value is $x=-7$.

Next, use the inequality $z<x$. Since $x=-7$, you have $z<-7$, so $z$ must be negative.

Finally, look at $y/x=z$. Because $x$ is negative and $z$ is negative, $y$ must be positive (a positive divided by a negative is negative).

So:

• $x$ is negative
• $y$ is positive

That’s enough to compare the quantities: Quantity B is greater than Quantity A.

Defined

On the GRE, you’ll most often see the word defined in two settings: functions and sequences (you’ll study both in more detail later). For now, focus on what “defined” tells you about which values are allowed.

Defined functions

The phrase is defined for means “the expression works for these inputs.” For example:

The function $f$ is defined for all real values of $x$ other than $1$ by $f(x)=(x-2)/(x-1)$

You can restate this as:

The function $f$ works for all values of $x$ other than $1$, where $f(x)=(x-2)/(x-1)$

Here, $x$ cannot be $1$ because the denominator $x-1$ would become $0$, making the expression undefined. That’s why the function is defined for all numbers other than $1$.

Defined sequences

When a sequence is defined by a rule, the rule tells you how its terms are determined. Together with any required starting value(s), it determines the sequence. Here’s an example:

A sequence is defined by the following equation: $S_n={S_{n-1}}^2$

If the first term in the sequence is $3$, what is the average of the next $3$ terms in the sequence?

This rule says each term ($S_n$) equals the previous term ($S_{n-1}$) squared.

A table helps you keep track of the terms. Here, $n$ is the term number (position), and $S_n$ is the value of that term.

$n$	$S_n$
$1$	$S_1=3$
$2$	$S_2=(S_1{)}^2=3^2=9$
$3$	$S_3=(S_2{)}^2=9^2=81$
$4$	$S_4=(S_3{)}^2=81^2=6561$

The next three terms after $3$ are $[9,81,6561]$. Their average is:

$$(9+81+6561)/3=2217$$

Common themes

• Questions that allow the possibility of undefined numbers often include trap answers. For example, if $0<x<5$, but $x$ cannot be $2$ because it makes a solution undefined, $2$ will very likely be a trap answer choice.
• These trap questions often involve fractions with a quadratic in the denominator.