GMAT Math : DSQ: Understanding functions

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Example Question #11 :Dsq: Understanding Functions

$f ^{-1} (A) = 27$

Evaluate.

Statement 1: The graph ofincludes the point (15, 27) .

Statement 2: f(15) = 27

Possible Answers:

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

BOTH statements TOGETHER are insufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

Both statements are equivalent to the statement $f ^{-1} (27) = 15$ . This is of no help to us.

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Example Question #12 :Dsq: Understanding Functions

Does f (x) have an inverse?

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

Statement 2: f(1) = f(-1)

Possible Answers:

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 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.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

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

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

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Example Question #13 :Dsq: Understanding Functions

$\left \lfloor x\right \rfloor$ is defined to be the greatest integer less than or equal to.

Evaluate $\left \lfloor A^{2} + B \right \rfloor$ .

Statement 1: 0<A < 1

Statement 2: 0< B < 1

Possible Answers:

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.

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.

Correct answer:

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: A,B = 0.5

$\left \lfloor A^{2} + B \right \rfloor = \left \lfloor 0.5^{2} + 0.5 \right \rfloor = \left \lfloor 0.25 + 0.5 \right \rfloor= \left \lfloor 0.75\right \rfloor = 0$

Example 2: A,B = 0.8

$\left \lfloor A^{2} + B \right \rfloor = \left \lfloor 0.8^{2} + 0.8 \right \rfloor = \left \lfloor 0.64 + 0.8 \right \rfloor= \left \lfloor 1.44\right \rfloor = 1$

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Example Question #14 :Dsq: Understanding Functions

Isan odd function?

Statement 1: It is a polynomial of degree 3.

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

Possible Answers:

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.

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Correct answer:

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 ofin its domain, g(-x)= -g(x) .

It is odd if and only if its graph is symmetrical to the origin, so Statement 2 proves thatis 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: $g(x) = x^{3}$

$g(-x) = (-x)^{3} = (-1)^{3} \cdot x^{3} = -1 x^{3} = -g(x)$

so this is odd.

Case 2: $g(x) = x^{3} -1$

$g(1) = 1^{3}-1 = 1-1 =0$

$g(-1) = (-1)^{3} -1= -1-1 = -2$

$g(1) \neq g(-1)$ , so the function is not odd.

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Example Question #15 :Dsq: Understanding Functions

Isan odd function?

Statement 1: For each positive, f (-x) = -f(x)

Statement 2: f(0) = 1

Possible Answers:

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 insufficient to answer the question.

Correct answer:

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

Explanation:

By definition, forto be odd, then it must hold that f (-x) = -f(x) for every value ofin its domain.

If f (-x) = -f(x) for every positive, as stated in Statement 1, then -f (-x) = f(x) for all positive. Equivalently, f(-x) = -f (x) for allnegative. But this does not give us any information about the behavior ofat 0, so the picture is incomplete.

But Statement 2 alone provesis not odd. This is because ifis odd, the definition forces f (0) = -f(-0) = -f(0) , which forces f(0) = 0 . Statement 2 contradicts this.

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Example Question #16 :Dsq: Understanding Functions

Isan even function, an odd function, or neither?

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

Statement 2: The graph ofis a line through the origin.

Possible Answers:

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 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.

Correct answer:

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 provesodd.

A function is odd if and only if, for eachin the domain, f(-x) = -f(x) . A linear function through the origin - that is, one with-intercept 0 - can be written as f(x) = mx for some; since

f(-x) = m (-x) = -mx = -f(x) ,

we knowis odd.

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例子问题# 17:Dsq: Understanding Functions

$\left \lceil x \right \rceil$ is defined as the least integer greater than or equal to.

Evaluate: $\left \lceil x^2 + y^2 \right \rceil$

Statement 1: $0 < x < \frac{1}{2}$

Statement 2: $0 < y < \frac{1}{2}$

Possible Answers:

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.

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.

Correct answer:

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 $\left \lceil x^2 + y^2 \right \rceil$ .

For example, if $x = \frac{1}{4 }$ and y = 1 ,

$\left \lceil x^2 + y^2 \right \rceil = \left \lceil \left ( \frac{1}{4} \right )^2 + 1^2 \right \rceil = \left \lceil1 \frac{1}{16} \right \rceil = 2$

If $x = \frac{1}{4 }$ and y = 2 ,

$\left \lceil x^2 + y^2 \right \rceil = \left \lceil \left ( \frac{1}{4} \right )^2 + 2^2 \right \rceil = \left \lceil4 \frac{1}{16} \right \rceil = 5$

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

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

$0 < x < \frac{1}{2} \Rightarrow 0 < x ^{2}< \frac{1}{4}$

$0 < y< \frac{1}{2} \Rightarrow 0 < y ^{2}< \frac{1}{4}$

$0=0+0 <x^{2}+y^{2}< \frac{1}{4} + \frac{1}{4} = \frac{1}{2 }< 1$

Since $0<x^{2}+y^{2}< 1$ ,

$\left \lceil x^2 + y^2 \right \rceil = 1$ .

We have a definitive answer.

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Example Question #18 :Dsq: Understanding Functions

Does the function f (x) have an inverse?

Statement 1: f (1) = 1

Statement 2: f (-1) = 1

Possible Answers:

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.

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.

Correct answer:

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映射到相同的值 f(x) . Neither statement is sufficient to prove or disprove this, Both statements together, however, demonstrate that there are two values of, (-1,1) , that map into the samevalue (1) , sohas no inverse.

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Example Question #19 :Dsq: Understanding Functions

$\left \lfloor N \right \rfloor$ is defined to be the greatest integer less than or equal to.

$\left \lceil N\right \rceil$ is defined to be the least integer greater than or equal to.

Is it true that M = N ?

(a) $\left \lfloor M \right \rfloor =\left \lfloor N \right \rfloor$

(b) $\left \lceil M\right \rceil = \left \lceil N\right \rceil$

Possible Answers:

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.

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.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

If M = N , then both statements are true. But both statements can also be true in some cases where $M \neq N$ .

For example, if M = 8.5, N= 8.6 , then $\left \lfloor M \right \rfloor =\left \lfloor N \right \rfloor = 8$ and $\left \lceil M\right \rceil = \left \lceil N\right \rceil = 9$ .

The two together are inconclusive.

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Example Question #20 :Dsq: Understanding Functions

$\left \lfloor N \right \rfloor$ is defined to be the greatest integer less than or equal to.

$\left \lceil N\right \rceil$ is defined to be the least integer greater than or equal to.

Isan integer?

Statement 1: $\left \lfloor N \right \rfloor = N$

Statement 2: $\left \lceil N \right \rceil = N$

Possible Answers:

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.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

Ifis not an integer, then $\left \lfloor N \right \rfloor < N$ - for example, $\left \lfloor 1.5 \right \rfloor = 1$ , so $\left \lfloor 1.5 \right \rfloor < 1.5$ . Therefore, by Statement 1 alone, since $\left \lfloor N \right \rfloor = N$ ,must be an integer. By a similar argument, Statement 2 alone provesis an integer.

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GMAT Math : DSQ: Understanding functions

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #11 :Dsq: Understanding Functions

Example Question #12 :Dsq: Understanding Functions

Example Question #13 :Dsq: Understanding Functions

Example Question #14 :Dsq: Understanding Functions

Example Question #15 :Dsq: Understanding Functions

Example Question #16 :Dsq: Understanding Functions

例子问题# 17:Dsq: Understanding Functions

Example Question #18 :Dsq: Understanding Functions

Example Question #19 :Dsq: Understanding Functions

Example Question #20 :Dsq: Understanding Functions

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