GMAT Math : Functions/Series

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Example Question #870 : Data Sufficiency Questions

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

Statement 1: Its domain is $\left \{ 1,3,5, 7, 9, 11\right \}$ .

Statement 2: The line x= 5 passes through its graph twice.

Possible Answers:

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.

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:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

If Statement 1 alone is assumed, then, since there are only six domain elements and ten points in the relation, at least one of the domain elements must match with more than one range element. This forces the relation to not be a function.

If Statement 2 alone is assumed, then, since x= 5 is a vertical line that passes through the graph twice, the relation fails the vertical line test and is therefore not a function.

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Example Question #41 : Dsq: Understanding Functions

Define and to be functions. Does $f \circ g$ have an inverse?

Statement 1: has an inverse.

Statement 2: has an inverse.

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.

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.

Correct answer:

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

Explanation:

$\left (f \circ g \right )(x) = f \left ( g\left ( x\right )\right )$ . This defintion will come into play here.

A function has an inverse if and only if it is "one-to-one" - that is, if

f(a) = f(b) if and only if a = b .

Assume statement 1 only.

$f \left ( g\left ( a\right )\right ) = f \left ( g\left ( b\right )\right )$ if and only if $g\left ( a\right ) =g(b)$ . However, since it is not known whether has an inverse, it is possible for $g\left ( a\right ) =g(b)$ with either a = b or $a \ne b$ . Transitively, it is possible for $f \left ( g\left ( a\right )\right ) = f \left ( g\left ( b\right )\right )$ with either a = b or $a \ne b$ .

Assume Statement 2 only.

$g\left ( a\right ) =g(b)$ if and only if a = b . But it is possible for $f \left ( g\left ( a\right )\right ) = f \left ( g\left ( b\right )\right )$ with $g\left ( a\right ) =g(b)$ or $g\left ( a\right ) \ne g(b)$ - and, subsequently, with a = b or $a \ne b$ . Transitively, it is possible for $f \left ( g\left ( a\right )\right ) = f \left ( g\left ( b\right )\right )$ with either a = b or $a \ne b$ .

Assume both statements are true. Then

$f \left ( g\left ( a\right )\right ) = f \left ( g\left ( b\right )\right )$ if and only if $g\left ( a\right ) =g(b)$ , and $g\left ( a\right ) =g(b)$ if and only if a = b . Transitively,

$f \left ( g\left ( a\right )\right ) = f \left ( g\left ( b\right )\right )$ if and only if a = b .

Therefore,

$\left (f \circ g \right )(a) = \left (f \circ g \right )(b)$ if and only if a = b , and, subsequently, $f \circ g$ has an inverse.

The two statements together - but neither alone - lead to an answer.

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Example Question #71 : Algebra

Define and to be functions on the real numbers. Does f-g have an inverse?

Statement 1: does not have an inverse.

Statement 2: does not have an inverse.

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 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:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

Examine these two examples.

Example 1: Let f(x) = g(x) = 1

Neither function has an inverse, since both functions pair all values with the same value, namely, 1.

(f-g)(x) = f(x) - g(x) = 1 - 1= 0

f-g does not have an inverse, since it pairs all values with the same value, namely, 0.

Therefore, and have no inverse, and f-g has no inverse.

Example 2: Let $f(x) = x^{2}+x$ and $g(x) = x^{2}$ .

has no inverse, since $f(0) = 0^{2}+0 = 0$ and $f(-1) =(-1) ^{2} +(-1) = 1-1 = 0$ - that is, pairs at least two values with the same value.

has no inverse, since $g (1) = 1^{2} = 1$ and $g(-1 ) = (-1)^{2} = 1$ ; that is, pairs at least two values with the same value.

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

f-g is the identity function, which has itself as an inverse.

This demonstrates that, if and do not have inverses, it is possible for f-g to have an inverse or to not have an inverse. Therefore, the two statements together are inconclusive.

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Example Question #43 : Dsq: Understanding Functions

Define and to be functions on the set of real numbers. Does f+g have an inverse?

Statement 1: has an inverse.

Statement 2: has an inverse.

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.

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.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

Examine these two examples.

Example 1: f(x) = 3x and g(x) = 2x .

f(x) = 3x can be proved to have an inverse as follows:

y= 3x

Switch the variables:

3y = x

$y = \frac{1}{3}x$

$f^{-1}(x) = \frac{1}{3}x$

Therefore, has an inverse; a smiliar proof shows that has an inverse $g^{-1} (x)= \frac{1}{2}x$ .

(f+g)(x) = 3x + 2x = 5x , which can be similarly be shown to have inverse

$(f+g)^{-1}(x) = \frac{1}{5}x$ .

Example 2: f(x) = 3x and g(x) = -3x .

Again, and can be shown to have inverses, $f^{-1}(x) = \frac{1}{3}x$ and $g^{-1}(x) =-\frac{1}{3}x$ .

. However,

(f+g)(x) = 3x + (-3x) = 0 .

This function has no inverse, since this function pairs multiple values of with the same value of , 0.

In both cases, both and have inverses, but in one case, f+g has an inverse, and in the other case, f+g does not. The two statements together are inconclusive.

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Example Question #41 : Dsq: Understanding Functions

Is a given relation a function?

Statement 1: The line x = 10 passes through its graph infinitely many times.

Statement 2: The line y= 10 passes through its graph infinitely many times.

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 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.

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:

If Statement 1 alone is assumed, then, since x= 10 is a vertical line that passes through the graph of the relation more than once, the relation fails the vertical line test, and the relation can be proved to not be a function.

However, there is no restriction on how many times a horizontal line such as y= 10 can pass through the graph of a relation for it to be or not to be a function. Statement 2 proves nothing either way.

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Example Question #45 : Dsq: Understanding Functions

Is a given relation a function?

Statement 1: Two of its ordered pairs have -coordinate .

Statement 2: Two of its ordered pairs have -coordinate .

Possible Answers:

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.

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:

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

Explanation:

Statement 1 alone disproves that the relation is a function, since in a function, no -coordinate can be paired with two -coordinates. However, statement 2 alone does not prove or disprove the relation to be a function, since it is permissible for two -coordinates to be paired with the same -coordinate.

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Example Question #46 : Dsq: Understanding Functions

The volume of a right circular cylinder is $\pi ^{x}$ ; the radius of its base is $\pi$ .

Give the height of the cylinder.

Possible Answers:

$\pi ^{\frac{1}{2}x- 1}$

$\pi ^{x-3}$

$\pi ^{\sqrt{x-2}}$

$\pi ^{\frac{1}{2}x}$

$\pi ^{\frac{1}{3}x }$

Correct answer:

$\pi ^{x-3}$

Explanation:

The volume of a right circular cylinder, given base with radius and given height , is

$V = \pi r^{2}h$ .

Setting $V = \pi ^{x}$ and $r = \pi$ :

$\pi ^{x} = \pi \cdot \pi^{2} \cdot h$

$\pi ^{x} = \pi^{2+1} \cdot h$

$\pi ^{x} = \pi^{3} \cdot h$

$\frac{ \pi ^{x}}{ \pi^{3}} =\frac{ \pi^{3} \cdot h }{ \pi^{3}}$

$h = \pi ^{x-3}$

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Example Question #47 : Dsq: Understanding Functions

The volume of a right circular cylinder is $\pi ^{x}$ ; its height is $\pi$ .

Give the radius of a base of the cylinder.

Possible Answers:

$\pi ^{x-3}$

$\pi ^{\sqrt{x-2}}$

$\pi ^{\frac{1}{3}x }$

$\pi ^{\frac{1}{2}x- 1}$

$\pi ^{\frac{1}{2}x}$

Correct answer:

$\pi ^{\frac{1}{2}x- 1}$

Explanation:

The volume of a right circular cylinder, given base with radius and given height , is

$V = \pi r^{2}h$ .

Setting $V = \pi ^{x}$ and $h= \pi$ :

$\pi ^{x} = \pi \cdot r ^{2} \cdot \pi$

$\pi ^{x} = \pi^{2} \cdot r ^{2}$

$\frac{ \pi ^{x}}{\pi^{2} } = \frac{\pi^{2} \cdot r ^{2}}{\pi^{2} }$

$r ^{2} = \pi ^{x-2}$

$\left (r ^{2} \right )^{\frac{1}{2}}= \left (\pi ^{x-2} \right )^{\frac{1}{2}}$

$r ^{2 \cdot \frac{1}{2}}= \pi ^{(x-2)\cdot \frac{1}{2}}$

$r = \pi ^{\frac{1}{2}x- 1}$

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Example Question #81 : Algebra

The first two terms of a sequence, $a_{1}$ and $a_{2}$ , are given specific values, which you are not given. Each successive term is equal to the sum of the two preceding terms. For example, the fifth term is equal to the sum of the third and fourth terms.

Calculate $a_{1}$ .

Statement 1: $a_{5} = 18$

Statement 2: $a_{8} = 76$

Possible Answers:

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 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.

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

Correct answer:

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

Explanation:

Suppose the first and second terms of the sequence are and , respectively. The first eight terms are

A + B

B + (A+B ) , or A + 2B

(A+B )+ (A+2B ) , or 2A + 3B - this is the fifth term.

Continuing to add:

3A+5B

5A + 8B

8A+13B - the eighth term.

From Statement 1 alone, we get that

2A + 3B = 18

From Statement 2 alone, we get that

8A+13B = 76

Both statements are equations in two variables, so neither statement alone tells us the value of either. But if we know both statements together, we have a system of equations that can be solved as follows:

2A + 3B = 18

-4(2A + 3B )= -4( 18)

-8A -12B = -72

$\underline{\; \; 8A+13B = 76}$

B = 4

2A + 3 (4) = 18

2A +12 = 18

2A = 6

A = 3 - this is $a_{1}$ .

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Example Question #82 : Algebra

Evaluate $a_{10}$ .

Statement 1: $a_{9} = 123$ and $a_{11} = 322$

Statement 2: The arithmetic mean of $a_{8}$ and $a_{9}$ is 99.5.

Possible Answers:

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.

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

EITHER STATEMENT ALONE provides sufficient information to answer the question.

Correct answer:

EITHER STATEMENT ALONE provides sufficient information to answer the question.

Explanation:

Assume Statement 1 alone. By the rule, $a_{9}+ a_{10} = a_{11}$ , so by substitution:

$123+ a_{10} =322$

$a_{10} =322 - 122 = 199$ .

Assume Statement 2 alone. The arithmetic mean of two numbers is half their sum, so

$\frac{a_{8}+a_{9}}{2} = 99.5$

$\frac{a_{8}+a_{9}}{2} \cdot 2 = 99.5 \cdot 2$

$a_{8}+a_{9} = 199$

By the rule, $a_{8}+a_{9} = a_{10}$ , so

$a_{10} = 199$ .

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GMAT Math : Functions/Series

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #870 : Data Sufficiency Questions

Example Question #41 : Dsq: Understanding Functions

Example Question #71 : Algebra

Example Question #43 : Dsq: Understanding Functions

Example Question #41 : Dsq: Understanding Functions

Example Question #45 : Dsq: Understanding Functions

Example Question #46 : Dsq: Understanding Functions

Example Question #47 : Dsq: Understanding Functions

Example Question #81 : Algebra

Example Question #82 : Algebra

All GMAT Math Resources

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