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.

Undefined

There are four main situations where you’ll get an undefined result:

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

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

2. The square root of any negative number is undefined.

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

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. Slope is defined as “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 or defined by often means “the expression works for these inputs.” For example:

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

You can restate this as:

The function $f$ works with all values of $x$ except $1$, 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, it means every term in the sequence is determined by that rule. 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.

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

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