# GMAT Math : DSQ: Understanding functions

## Example Questions

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### Example Question #1 : Dsq: Understanding Functions

Define  .

Evaluate .

Statement 1:

Statement 2:

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.

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

Explanation:

If you know that , then you can calculate:

Knowing the -values of  that are paired with -value 9, however, is neither necessary nor useful here.

### Example Question #1 : Dsq: Understanding Functions

The first term of an arithmetic sequence is 100. What is the second term?

Statement 1: The sum of the third and fourth terms is 320.

Statement 2: The tenth term subtracted from the twelfth term yields a difference of 48.

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.

EITHER 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.

Explanation:

Let  be the common difference of the sequence. Then the third and fourth terms are, respectively,  and  . If their sum is 320, then

If the difference between the twelfth term  and tenth term  is 24, then

From either statement, the common difference can be calculated, then added to 100 to get the second term, 124.

### Example Question #1 : Dsq: Understanding Functions

What is the first term of a geometric sequence?

Statement 1: The product of the second and third terms is 4,096.

Statement 2: The product of the first and fourth terms is 4,096.

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.

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.

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

Let  be the first term and  be the common ratio. Then the first four terms are:

The two statements are equivalent to the equations  and . But they turn out to be equivalent as can be seen here:

This (common) statement alone is not enough to allow us to calculate .

### Example Question #1 : Dsq: Understanding Functions

This relation has five different ordered pairs: is it a function?

Statement 1:

Statement 2:

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.

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.

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

Explanation:

To prove that a relation is a function, you must prove that no -coordinate is matched with more than one -coordinate. Statement 2 proves that this is false, since 5 is now matched with both  and , which are different numbers regardless of . Statement 1 is irrelevant, since it does not prove or disprove this condition.

### Example Question #5 : Dsq: Understanding Functions

Evaluate .

Statement 1: The graph of  includes the point .

Statement 2:

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

Explanation:

If , then , so this question is equivalent to evaluating

If the ordered pair  is on the graph of , then , so .

Knowing that  is of no help, as this just tells us .

### Example Question #6 : Dsq: Understanding Functions

What is the first term of the geometric sequence?

Statement 1: The sum of the second and third terms is 90.

Statement 2: The sum of the third and fourth terms is 450.

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.

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.

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

Explanation:

Let  be the first term and  be the common ratio. Then the first four terms are:

.

The two statements below are equivalent to  and , respectively. Neither, alone, will help you figure out  or . If you know both, you can use algebra to deduce their values:

Divide both sides of the first equation by both sides of the second:

Substitute this value into either equation. We'll use the 2nd equation:

### Example Question #7 : Dsq: Understanding Functions

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

Evaluate

Statement 1:

Statement 2:

BOTH statements TOGETHER are insufficient 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.

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 is not enough to answer this question.

Case 1:

Then

Case 2:

Then

If Statement 2 is true, however:

### Example Question #8 : Dsq: Understanding Functions

This relation has the following five ordered pairs: is it a function?

Statement 1:

Statement 2:

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.

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

Explanation:

To prove that a relation is a function, you must prove that no -coordinate is matched with more than one -coordinate. Statement 1 proves that this is true. Statement 2 is irrelevant; it is possible for more than one -coordinate to be matched with the same -coordinate in a function.

### Example Question #1 : Dsq: Understanding Functions

This relation has five different ordered pairs: is it a function?

Statement 1:

Statement 2:

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.

BOTH statements TOGETHER are insufficient 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.

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

Explanation:

To prove that a relation is a function, you must prove that no -coordinate is matched with more than one -coordinate. Statement 1 proves that this is false, since 3 is now matched with both 7 and 9. Statement 2 is irrelevant, since it does not prove or disprove this condition.

### Example Question #831 : Data Sufficiency Questions

Define an operation  as follows:

For any real numbers ,

Evaluate