GMAT Math : Algebra

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Example Questions

Example Question #61 : Algebra

Given two functions and defined on the set of real numbers, which, if either, is the greater quantity, $\left ( f \circ g \right )(0 )$ or $\left ( g\circ f \right )(0 )$ ?

Statement 1: f (0 ) = g (0)

Statement 2: f (x)= g (x) + x for all real values of .

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 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.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

Statement 1 is a consequence of Statement 2, so we need only show that Statement 2 provides insufficient information.

Let N = f (0 ) = g (0) . Then

$\left ( f \circ g \right )(0 ) = f(g(0)) = f (N) = g(N)+N$

and

$\left (g \circ f \right )(0 ) = g(f(0)) = g (N)$ .

However, without knowing the sign of , we cannot determine whether f(N) or g(N) is the greater quantity.

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Example Question #22 : Dsq: Understanding Functions

Define an operation $\Sigma$ on two real numbers as follows:

For all real numbers a,b ,

$a \Sigma b = 10 - a + b$ .

True or false: $M\Sigma N=N \Sigma M$

Statement 1: M - N = 1

Statement 2: M + N = 7

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.

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:

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

Explanation:

$M\Sigma N= 10 - M + N$

Similarly, $N\Sigma M= 10 - N + M$ .

Therefore, for $M\Sigma N=N \Sigma M$ , it must hold that

10 - M + N = 10 - N + M .

10 - M + N - 10 + M + N= 10 - N + M - 10 + M + N

2N= 2 M

N = M

M-N = 0

Statement 1 contradicts this; Statement 2 neither confirms nor contradict this.

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Example Question #62 : Algebra

Define an operation $\bigstar$ on two real numbers as follows:

$a \bigstar b = a\sqrt{\left |b \right |}$

Is $M \bigstar N$ positive, negative, or zero?

Statement 1: M < 0

Statement 2: N< 0

Possible Answers:

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.

Correct answer:

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

Explanation:

$\sqrt{\left | N\right |}$ is positive or zero, so if Statement 1 is assumed, and is negative, then $M \bigstar N = M \sqrt{\left | N \right |}$ , the product of a negative number and a nonnegative number, must be nonpositive. It can be negative or zero, however.

If Statement 2 is assumed, the expression can be of any sign, since, although $\sqrt{\left | N\right |}$ must be positive, no information is given about the sign of .

If both statements are assumed, then since is nonzero, $\sqrt{\left | N\right |}$ is positive; also, is negative. $M \bigstar N = M \sqrt{\left | N \right |}$ is the product of two numbers of unlike sign, and therefore, it can be determined that it is a negative number.

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Example Question #31 : Dsq: Understanding Functions

John is writing out an arithmetic sequence. How many terms does he need to write out before he writes a term greater than or equal to 1,000?

Statement 1: The fourth term is 50.

Statement 2: The twentieth term is 418.

Possible Answers:

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 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:

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

Explanation:

Knowing one term of a sequence will not help you find any other terms, so neither statement alone will answer the question. But knowing two terms and knowing that the sequence is arithmetic will allow you to find the common difference .

The twentieth term is 16d greater than the fourth term, so take the difference and divide by 16:

$d = \frac{418-50}{16} = 23$

Now solve this inequality for to find the minimum number of terms needed to exceed 1,000:

$50 + 23t \geq 1,000$

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Example Question #63 : Algebra

Define an operation $\bullet$ on the real numbers as follows:

$a \bullet b = (a^{2} +1) (b- 10)$

Is $a \bullet b$ positive, negative, or zero?

Statement 1: a > 0

Statement 2: b > 0

Possible Answers:

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.

EITHER statement ALONE is sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

$a \bullet b = (a^{2} +1) (b- 10)$ , and $a ^{2} + 1 \geq 0 + 1 = 1 > 0$ . Therefore, since $a ^{2} + 1$ is a positive number, the sign of b - 10 is the sign of $a \bullet b$ . This makes Statement 1 neither necessary nor helpful. We need to know whether b - 10 is greater than, equal to, or less than 0, or, equivalently, whether is greater than, equal to, or less than 10. Statement 2 does not tell us this either.

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Example Question #861 : Data Sufficiency Questions

What is f(g(1)) ?

(1) f(x) = 3x - 1

(2) g(x) = $(f(x))^{2}$

Possible Answers:

A: Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient

E: Statements (1) and (2) TOGETHER are not sufficient

C: BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient

B: Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient

D: EACH statement ALONE is sufficient

Correct answer:

C: BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient

Explanation:

Statement (1) does not give us any information about g(x), so it is not sufficient.

Statement (2) alone gives us the relationship between the functions f and g but does not give us any information about f(x), so it is not sufficient.

Both statement together allow us to get an expression for both f and g:

f(x) = 3x - 1

$g(x)=(f(x))^{2}=(3x-1)^{2}=9x^{2}-6x+1$

With an expression for both functions we can estimate f(g(1)):

g(1)=9-6+1=4 and $f(4)=3\cdot4-1=11$

So the correct answer is C.

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Example Question #31 : Dsq: Understanding Functions

Does a function have an inverse?

Statement 1: f(4)- f(-4)= 0

Statement 2: f(7) +f(-7)= 0

Possible Answers:

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.

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:

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

Explanation:

A necessary and sufficient condition for not to have an inverse is for f(a) = f(b) for distinct a,b in the domain of .

If f(4)- f(-4)= 0 , then f(4)=f(-4) , meaning that pairs at least two domain values, 4 and , with the same range value. Therefore, from Statement 1 alone, it follows that does not have an inverse.

If f(7) +f(-7)= 0 , then f(7) = -f(-7) , meaning pairs two different domain values with two different range values. This is nt a contradiction of the conditions for an inverse to not exist, but this does not prove that an inverse does exist. Statement 2 is unhelpful either way.

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Example Question #863 : Data Sufficiency Questions

Does a function have an inverse?

Statement 1: Every vertical line passes through the graph of exactly once.

Statement 2: Every horizontal line passes through the graph of exactly once.

Possible Answers:

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.

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

Statement 1 is simply an assertion that the relation passes the vertical line test and is therefore a function. Since we know already that the relation is a function, this statement is unhelpful.

If Statement 2 is assumed, it follows that the graph passes the horizontal line test for the existence of an inverse. Statement 2 alone answers the question.

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Example Question #861 : Data Sufficiency Questions

A relation comprises ten ordered pairs. Is it a function?

Statement 1: The domain of the relation is $\left \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \right \}$ .

Statement 2: The range of the relation is $\left \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \right \}$ .

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.

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.

Correct answer:

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

Explanation:

The relation comprises ten ordered pairs. If Statement 1 alone is known, then the domain comprises ten elements, each of which must appear in exactly one ordered pair. Therefore, no domain element is matched with more than one range element, and the relation is a function.

If Statement 2 alone is known, then the range comprises ten elements, each of which must appear in exactly one ordered pair. But nothing is known about the domain. If no domain element is repeated among the ordered pairs, the relation is a function; otherwise, the relation is not a function.

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Example Question #865 : Data Sufficiency Questions

True or false: (fg)(1) < 0

Statement 1: f(M) < 0 for all real values of .

Statement 2: $g(x) = -x^{2}$

Possible Answers:

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.

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.

Correct answer:

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

Explanation:

$(fg)(1) = f(1) \cdot g(1)$ . So, for (fg)(1) to be negative, f(1) and g(1) must be of unlike sign.

From Statement 1 it can be determined that f(1) is negative, but no information about the sign of g(1) can be determined.

From Statement 2, it can be determined that $g(1) = -1^{2} = -1$ and is therefore negative, but no information about the sign of f(1) can be determined.

From the two statements together, both f(1) and g(1) can be proved negative, so their product, $f(1) \cdot g(1) = (fg)(1)$ , is positive.

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GMAT Math : Algebra

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #61 : Algebra

Example Question #22 : Dsq: Understanding Functions

Example Question #62 : Algebra

Example Question #31 : Dsq: Understanding Functions

Example Question #63 : Algebra

Example Question #861 : Data Sufficiency Questions

Example Question #31 : Dsq: Understanding Functions

Example Question #863 : Data Sufficiency Questions

Example Question #861 : Data Sufficiency Questions

Example Question #865 : Data Sufficiency Questions

All GMAT Math Resources

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