# GMAT Math : DSQ: Understanding functions

## Example Questions

### Example Question #21 : Functions/Series

Define

Is  greater than, less than, or equal to    ?

Statement 1:  is a positive number.

Statement 2:  is a negative number.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

One must have information about both  and  to answer the question, so neither statement is sufficient by itself.

Now assume both statements to be true. Then,  being positive and  being negative, .

### Example Question #22 : Dsq: Understanding Functions

Define

True or false: .

Statement 1:  is a positive number

Statement 2:  is a negative number

EITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

Assume both statements. Then  and . But this does not answer the question. For example,

If , then

making  true.

But if , then

making   false.

### Example Question #21 : Dsq: Understanding Functions

Define an operation  on two real numbers as follows:

Is  positive, negative, or zero?

Statement 1:

Statement 2:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Explanation:

Statement 1 alone does not provide a definite answer:

Case 1: If  and , then , and

Case 2:  If  and , then , and

However, if we assume Statement 2, then, since , we can see:

, and we know this is positive.

### Example Question #21 : Dsq: Understanding Functions

Define an operation  on two real numbers as follows:

Is  positive, negative, or zero?

Statement 1:  is negative.

Statement 2:  is positive.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Explanation:

If we know Statement 1 only - that  is negative - then, since  must be nonnegative,

must be a negative number minus a nonnegative number. This makes  negative.

If we know Statement 2 only - that  is positive - then we do not have a definite answer. For example,

but

### Example Question #24 : Dsq: Understanding Functions

What is the measure of   ?

Statement 1:  is complementary to an angle that measures .

Statement 2:  is supplementary to an angle that measures .

BOTH statements TOGETHER are insufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Explanation:

Complementary angles have degree measures that total , so the measure of an angle complementary to a  angle would have measure .

Supplementary angles have degree measures that total , so the measure of an angle supplementary to a  angle would have measure

From either statement alone, we know that .

### Example Question #2961 : Gmat Quantitative Reasoning

is defined as the greatest integer less than or equal to .

You are given that .

True or false:

Statement 1:  is an integer.

Statement 2:  is an integer.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

If  and  is an integer, then ; likewise for .

We show that the two statements together are not enough to draw a conclusion by assuming both statements are true and taking two cases:

Case 1:

is a false statement.

Case 2:

is a true statement.

### Example Question #27 : Dsq: Understanding Functions

is defined as the greatest integer less than or equal to .

You are given that .

True or false:

Statement 1:  is not an integer.

Statement 2: is an integer.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

Assume both statements to be true. We can show at least one case where  is true and one in which  is false.

Case 1:

Then  and .

This makes the two expressions equal and  a false statement.

Case 2:

Then  and .

This makes  true.

### Example Question #22 : Dsq: Understanding Functions

Gary is writing out a geometric sequence, in order.  How many terms does he need to write out before he writes a term greater than or equal to 1,000?

Statement 1: The sum of the first two terms is 5.

Statement 2: The sum of the third and fourth terms is 80.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

From Statement 1 alone, it cannot be determined what the first two terms or the common ratio are; the terms, for example, could be  or , giving a different common ratio in each case. By a smiliar argument Statement 2 alone is also insufficient.

If both statements are assumed, though, the following can be deduced. If  is the common ratio and  is the first term, then the first four terms are

.

Divide:

, so .

Knowing the common ratio and the first term allows us to determine all of the numbers of the sequence and to find the first one greater than or at least 1,000.

### Example Question #25 : Dsq: Understanding Functions

Given two functions  and  defined on the set of real numbers, which, if either, is the greater quantity,  or  ?

Statement 1:

Statement 2:  for all real values of .

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

Statement 1 is a consequence of Statement 2, so we need only show that Statement 2 provides insufficient information.

Let . Then

and

.

However, without knowing the sign of , we cannot determine whether  or  is the greater quantity.

### Example Question #22 : Functions/Series

Define an operation  on two real numbers as follows:

For all real numbers ,

.

True or false:

Statement 1:

Statement 2:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Explanation:

Similarly, .

Therefore, for , it must hold that

.