GMAT Math : Functions/Series

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Example Question #11 : Dsq: Understanding Functions

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

Evaluate .

Statement 1: The graph of includes 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 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.

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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

Is an 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 of in its domain, g(-x)= -g(x) .

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: $g(x) = x^{3}$

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

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

Is an 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, for to be odd, then it must hold that f (-x) = -f(x) for every value of in 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 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 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

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.

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 proves odd.

A function is odd if and only if, for each in 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 know is odd.

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Example Question #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 #11 : Dsq: Understanding Functions

Does the function f (x) have an inverse?

Statement 1: f (1) = 1

Statement 2: f (-1) = 1

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.

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.

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 map into the same value of 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 same value (1) , so has 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 .

Is an 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:

If is 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 proves is an integer.

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GMAT Math : Functions/Series

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

Example Question #17 : Dsq: Understanding Functions

Example Question #11 : Dsq: Understanding Functions

Example Question #19 : Dsq: Understanding Functions

Example Question #20 : Dsq: Understanding Functions

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