GMAT Math : DSQ: Understanding functions

Example Questions

Example Question #11 : Dsq: Understanding Functions

Evaluate .

Statement 1: The graph of  includes the point .

Statement 2:

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.

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 insufficient to answer the question.

Explanation:

Both statements are equivalent to the statement . This is of no help to us.

Example Question #41 : Algebra

Does   have an inverse?

Statement 1: There exists only one horizontal line that intersects the graph of more than once.

Statement 2:

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 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:

Statement 1 is enough to disprove that has an inverse; fails the horizotal line test, which states that for to have an inverse, no horizontal line can intersect its graph more than once.

Statement 2 is also enough to disprove that has an inverse, since for to have an inverse, no more than one -coordinate can be matched with the same -coordinate.

Example Question #41 : Algebra

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

Evaluate .

Statement 1:

Statement 2:

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:

Even if you know both statements, you cannot answer this question with certainty.

Example 1:

Example 2:

Example Question #11 : Dsq: Understanding Functions

Is  an odd function?

Statement 1: It is a polynomial of degree 3.

Statement 2: Its graph is symmetrical with respect to the origin.

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.

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:

is odd if and only if, for any value of  in its domain, .

It is odd if and only if its graph is symmetrical to the origin, so Statement 2 proves that  is an odd function.

Statement 1 does not provide enough information, however; we can give at least one third-degree polynomial that is odd and one that is not:

Case 1:

so this is odd.

Case 2:

, so the function is not odd.

Example Question #12 : Dsq: Understanding Functions

Is  an odd function?

Statement 1: For each positive  ,

Statement 2:

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.

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

Explanation:

By definition, for  to be odd, then it must hold that   for every value of  in its domain.

If  for every positive , as stated in Statement 1, then    for all positive . Equivalently,   for all negative . But this does not give us any information about the behavior of  at 0, so the picture is incomplete.

But Statement 2 alone proves  is not odd. This is because if  is odd, the definition forces , which forces . Statement 2 contradicts this.

Example Question #11 : Dsq: Understanding Functions

Is  an even function, an odd function, or neither?

Statement 1: The graph of  is symmetric with respect to the origin.

Statement 2: The graph of  is a line through the origin.

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.

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.

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.

Explanation:

A function is odd if and only if its graph has symmetry with respect to the origin, so Statement 1 proves  odd.

A function is odd if and only if, for each  in the domain, . A linear function through the origin - that is, one with -intercept 0 - can be written as  for some ; since

,

we know  is odd.

Example Question #2951 : Gmat Quantitative Reasoning

is defined as the least integer greater 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.

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 sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

Statement 1 is insufficient to calculate .

For example, if  and ,

If  and ,

A reciprocal argument can be used to show Statement 2 is also insufficient.

From both statements together, however, we know the following:

Since  ,

.

Example Question #12 : Functions/Series

Does the function  have an inverse?

Statement 1:

Statement 2:

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.

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:

has an inverse if and only if no two values of  map into the same value of . Neither statement is sufficient to prove or disprove this, Both statements together, however, demonstrate that there are two values of , , that map into the same  value , so  has no inverse.

Example Question #18 : Dsq: Understanding Functions

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

is defined to be the least integer greater than or equal to .

Is it true that  ?

(a)

(b)

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.

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 insufficient to answer the question.

Explanation:

If , then both statements are true. But both statements can also be true in some cases where .

For example, if , then  and .

The two together are inconclusive.

Example Question #13 : Dsq: Understanding Functions

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

is defined to be the least integer greater than or equal to .

Is  an integer?

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.

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.