# GMAT Math : DSQ: Understanding functions

## Example Questions

### Example Question #81 : Algebra

True or false:  is an arithmetic sequence.

Statement 1:

Statement 2:

BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

EITHER STATEMENT ALONE provides sufficient information to answer the question.

BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

Explanation:

An arithmetic sequence is one in which the difference of each term in the sequence and the one preceding is constant.

The two statements together demonstrate that two such differences are equal to each other, and that two other such differences are equal to each other. No information, however, is given about any other differences, of which there are infinitely many. Therefore, the question of whether the sequence is arithmetic or not is unresolved.

### Example Question #81 : Algebra

Give the first term of an arithmetic sequence .

Statement 1: The arithmetic mean of  and  is 24.

Statement 2: The common difference of the sequence is 10.

STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.

BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

EITHER STATEMENT ALONE provides sufficient information to answer the question.

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

Explanation:

Each term of an arithmetic sequence is the preceding term plus the same number, the common difference.

Assume Statement 1 alone, and examine the sequences:

Both sequences are arithmetic; the first has common difference 8, the second, common difference 9. In both sequences, the arithmetic mean of the second and third terms - half their sum - is . However, the first term differs.

Assume Statement 2 alone. The common difference alone is not enough to determine the first term, as evidenced in these two sequences:

both of which have common difference 10.

Now assume both statements. The arithmetic mean of  and  is 24, so

or

Also, the common difference is 10, so

These two equations form a two-by-two linear system which can be solved as follows:

### Example Question #51 : Dsq: Understanding Functions

Above is the graph of a function .

Given: a function  with domain

True or false:  exists.

Statement 1: For each  such that , it holds that  .

Statement 2: For each  such that , it holds that  .

STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.

EITHER STATEMENT ALONE provides sufficient information to answer the question.

BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.

Explanation:

in each statement, so the graph of the function  is the same as that of the function  translated three units right. However, we are restricting the domain of  to . Each of the two statements examines one half of the graph. See the graph below, which divides the graph into the portion on the domain  (in blue; discussed in Statement 1) and the portion on the domain  (in green, discussed in Statement 2):

Assume Statement 1 alone. The portion of the graph of  on the domain  passes the horizontal line test, since no horizontal line passes through it twice. However, without knowing anything about the other half of the graph, the question about whether  exists cannot be resolved.

Assume Statement 2 alone. Notice that we can draw a horizontal line through this portion of the graph that passes through it twice - would work. This half of the graph alone proves that  does not exist.

### Example Question #85 : Algebra

Give the first term of an arithmetic sequence

Statement 1: The eighth and ninth terms are 65 and 72, respectively.

Statement 2:

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

EITHER STATEMENT ALONE provides sufficient information to answer the question.

STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.

STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.

Explanation:

From Statement 1, the difference of two consecutive terms  and  is ; since, in an arithmetic sequence,

we can substitute  and find .

However, Statement 2 alone gives insufficient information helpful in finding ; for example, the sequences

and

have the characteristic that  - but  differs between them.

### Example Question #51 : Dsq: Understanding Functions

Given a function , it is known that:

Does  have an inverse?

Statement 1: The range of  is the set .

Statement 2: The domain of  is the set .

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

EITHER STATEMENT ALONE provides sufficient information to answer the question.

BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.

Explanation:

A function  has an inverse if and only if, if , then , or, equivalently, if , then .

Assume Statement 2 alone. If  is the entire domain, then  cannot exist for any value of  not in that set. Also, it can be seen that for each  such that . Therefore,  has an inverse.

Assume Statement 1 alone. We show that the question of whether  has an inverse cannot be answered by taking two cases.

Case 1:  is the entire domain. If this is true, then the range is , and the situation described in Statement 2 exists; consequently,  has an inverse.

Case 2:  is the domain, and . The range is still the set . However, , so there exists  in the range such that , but . This means that  does not have an inverse.

### Example Question #88 : Algebra

Given a function , it is known that:

Does  have an inverse?

Statement 1:

Statement 2:

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.

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:

A function  has an inverse if and only if, if , then , or, equivalently, if , then .

If Statement 1 alone is assumed, then this condition is known to not be true, since  . Therefore,  does not have an inverse.

If Statement 2 alone is assumed, since no two values  and  are known such that  and , it is possible for  to have an inverse. However, there may or may not be other values in the domain of , any of which may be paired with range elements in the set . Therefore, Statement 2 does not resolve the issue of whether  has an inverse.

### Example Question #51 : Dsq: Understanding Functions

Evaluate .

Statement 1:  is an even function.

Statement 2:  is an odd function.

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.

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.

Explanation:

Assume Statement 1 alone. Since  is even, then by definition, for each  in its domain,

Specifically,

and

.

Without further information, this expression cannot be evaluated.

Assume Statement 2 alone. Since  is odd, then by definition, for each  in its domain,

.

Specifically,

and

.

### Example Question #51 : Dsq: Understanding Functions

Let  and  be functions, the doimains of both of which are the set of all real numbers. Is the function  odd, even, or neither?

Statement 1:  is neither odd nor even.

Statement 2:  is neither odd nor even.

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.

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:

Assume both statements are true. A function  is odd if, for all  in its domain, , and even if, for all  in its domain, . We show that knowing that neither  nor  is odd or even is insufficient to answer the question of whether  is odd, even, or neither by examining two scenarios.

Case 1:  and .

Since there exists at least one value  for which neither  nor  is neither odd nor even.

By a similar argument,  can be shown to be neither odd nor even.

However,

and, for all  in the domain,

,

making  even.

Case 2:  and .

Again,  is neither even nor odd, and  can be similarly demonstrated to be neither as well.

Since there is at least one value  in the domain of  such that  is neither odd nor even.

### Example Question #51 : Dsq: Understanding Functions

Evaluate .

Statement 1:  is an odd function.

Statement 2:  is an odd function.

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.

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.

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:

, so we need to find the values of both  and  in order to answer this question.

Assume Statement 1 alone. By defintion of an odd function, from Statement 2, for every  in the domain of . In specific, setting

.

The only number whose opposite is itself is 0, so

and it follows that

.

However, we have no way of knowing the value of , so the expression cannot be evaluated.

By a similar argument, if Statement 2 alone is assumed, , but, since  is unknown, the expression cannot be evaluated.

Now assume both statements. It follows that , and

.

### Example Question #54 : Functions/Series

Given a function , it is known that:

Given a function , evaluate .

Statement 1:

Statement 2:  is an odd function.

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.

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.