GMAT Math : Algebra

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Example Questions

Example Question #161 : Algebra

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

Solve for : $\left \lfloor 3y - 1 \right \rfloor = 7$

Possible Answers:

$\left [ \frac{7}{3}, \frac{8}{3} \right )$

$\left [ \frac{8}{3}, 3 \right )$

$\left ( \frac{8}{3}, 3 \right )$

$\left (\frac{7}{3}, \frac{8}{3} \right ]$

$\left (\frac{8}{3}, 3 \right ]$

Correct answer:

$\left [ \frac{8}{3}, 3 \right )$

Explanation:

$\left \lfloor 3y - 1 \right \rfloor = 7$

means that the greatest integer less than or equal to 3y - 1 is 7. The equivalent statement is

$7 \leq 3y - 1 < 8$ .

Solve for as follows:

$7 + 1 \leq 3y - 1 + 1 < 8 + 1$

$8 \leq 3y < 9$

$8 \div 3 \leq 3y \div 3 < 9 \div 3$

$\frac{8}{3} \leq y< 3$

or, in interval form, $\left [ \frac{8}{3}, 3 \right )$

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Example Question #162 : Algebra

Which of the following pairs of statements is sufficient to prove that f(x) does not have an inverse?

Possible Answers:

$f(5) = 7,f(-5) = \frac{1}{7}$

None of these pairs of statements would be sufficient to prove that does not have an inverse.

f(1) = 10, f (-1) = -10

f (x) is not defined for x = 2 , f (x) is not defined for x = -2 ,

f(4) = 3,f(-4) = 3

Correct answer:

f(4) = 3,f(-4) = 3

Explanation:

For a function to have an inverse, no -coordinate can be paired with more than one -coordinate. Of our choices, only

f(4) = 3,f(-4) = 3

causes this to happen.

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Example Question #21 : Understanding Functions

Which of these functions is an example of a function with an inverse?

Possible Answers:

$g(x) = x^{4} - 1$

$f(x) = 3x^{2} + 5$

h(x) = 6x + 7

$C(x) = \cos 2 x$

$R(x) = \frac{1}{x^ {2}}$

Correct answer:

h(x) = 6x + 7

Explanation:

For a function to have an inverse, if f(a) = f(b) , then a = b . We can show this is not the case for four of these functions by providing one counterexample in each case.

$C(x) = \cos 2 x$

$C(0) = \cos 2 \cdot 0 = \cos 0 = 1$

$C( \pi ) = \cos 2 \cdot \pi = \cos 2 \pi = 1$

$f(x) = 3x^{2} + 5$

$f(1) = 3\cdot 1^{2} + 5 = 3\cdot 1 + 5= 3 + 5= 8$

$f(-1) = 3\cdot \left (-1 \right )^{2} + 5 = 3\cdot 1 + 5= 3 + 5= 8$

$g(x) = x^{4} - 1$

$g(1) = 1^{4} - 1 = 1-1 = 0$

$g(-1) = \left (-1 \right )^{4} - 1 = 1-1 = 0$

$R(x) = \frac{1}{x^ {2}}$

$R(1) = \frac{1}{1^ {2}} = \frac{1}{1} = 1$

$R(-1) = \frac{1}{\left (-1 \right )^ {2}} = \frac{1}{1} = 1$

However, we can show that h(x) = 6x + 7 has an inverse function by demonstrating that if h(a) = h(b) , then a = b .

h(a) = h(b)

6a + 7 = 6b + 7

6a + 7 -7 = 6b + 7-7

6a = 6b

$6a \div 6 = 6b \div 6$

a = b

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Example Question #21 : Understanding Functions

Define g(x) = 2x - |x| .

Which of the following would be a valid alternative way of expressing the definition of ?

Possible Answers:

g (x) = 3 x

$g (x) = \left\{\begin{matrix} 3x & \textrm{ if } x < 0\\ x& \textrm{ if } x \geq 0 \end{matrix}\right.$

$g (x) = \left\{\begin{matrix} 3x & \textrm{ if } x < 0\\ 2x& \textrm{ if } x \geq 0 \end{matrix}\right.$

$g (x) = \left\{\begin{matrix} 2x & \textrm{ if } x < 0\\ 3x& \textrm{ if } x \geq 0 \end{matrix}\right.$

$g (x) = \left\{\begin{matrix} x & \textrm{ if } x < 0\\ 3x& \textrm{ if } x \geq 0 \end{matrix}\right.$

Correct answer:

$g (x) = \left\{\begin{matrix} 3x & \textrm{ if } x < 0\\ x& \textrm{ if } x \geq 0 \end{matrix}\right.$

Explanation:

If x < 0 , then |x| = -x ,and subsequently, g(x) = 2x - |x| = 2x - (-x) = 2x + x = 3x

If $x \geq 0$ , then |x| = x ,and subsequently, g(x) = 2x - |x| = 2x - x = x

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Example Question #22 : Understanding Functions

Which of these functions is neither an even function nor an odd function?

Possible Answers:

$P(x) = x^{4} + 2x^{2} + 3$

$F(x) = x^{3} +4x$

$f(x) = x^{2}\cos x$

$G(x) = x^{3} + x^{2} + x + 1$

$g(x) = x ^{2}\sin x + \sin ^{3} x$

Correct answer:

$G(x) = x^{3} + x^{2} + x + 1$

Explanation:

A function is odd if and only if f(-x) = -f(x) for each value of in the domain; it is even if and only if f(-x) = f(x) for each value of in the domain. We can identify each of these functions as even, odd, or neither by evaluating each at and comparing the resulting expression to the definition of the function.

$f(x) = x^{2}\cos x$

$f(-x) = (-x)^{2}\cos (-x) = x^{2} \cos x = f(x)$

making even.

$g(x) = x ^{2}\sin x + \sin ^{3} x$

$g(-x) =\left ( -x \right ) ^{2} \sin \left \( -x \right ) + \sin ^{3} \left ( -x \right )$

$=x ^{2} \left ( -\sin x \right )+\left ( -\sin x \right )^{3}$

$=- x ^{2} \sin x -\sin^{3} x$

$= - \left ( x ^{2}\sin x + \sin ^{3} x \right ) = -g(x)$

making odd.

$F(x) = x^{3} +4x$

$F(-x) = (-x)^{3} +4 (-x)$

$= -x^{3} - 4x$

$= -\left ( x^{3} +4x \right ) = - F(x)$

making odd.

$P(x) = x^{4} + 2x^{2} + 3$

$P(-x) = (-x) ^{4} + 2(-x) ^{2} + 3$

$= x ^{4} + 2x ^{2} + 3 = P(x)$

making even.

$G(x) = x^{3} + x^{2} + x + 1$

$G(-x) = (-x)^{3} + (-x)^{2} +(-x) + 1 = - x^{3} + x^{2} - x + 1$

Since neither G(-x) = G(x) nor G(-x) = -G(x) , is neither even nor odd, making this the correct choice.

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Example Question #23 : Understanding Functions

If $x\ \bigstar\ y = x*y^2$ , then what does $(2\ \bigstar\ 2)\ \bigstar\ 2$ equal?

Possible Answers:

2^6

2^7

2^3

2^5

2^4

Correct answer:

2^5

Explanation:

This problem can be evaluated by determining what the value of the parentheses is and then using that to evaluate the rest of the term.

$(2\bigstar 2)=2*2^2=2^3$

The term then reduces to $2^3\bigstar 2 = 2^3*2^2=2^5$

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Example Question #163 : Algebra

Find the next term in the series $1,\frac{-1}{2},\frac{1}{4},\frac{-1}{8}.$

Possible Answers:

$\frac{1}{12}$

$-\frac{1}{12}$

$-\frac{1}{16}$

$\frac{1}{16}$

Correct answer:

$\frac{1}{16}$

Explanation:

We can see that we have a geometric series. The geometric factor can be found by dividing the second term by the first. Doing this, we get $\frac{-\frac{1}{2}}{1}=-\frac{1}{2}.$ To find the next term in the series, we simply multiply the last term by the geometric factor to get $\frac{-1}{8}*\frac{-1}{2}=\frac{1}{16}$ .

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Example Question #24 : Understanding Functions

Solve for f(g(4)) .

f(x)=x^2

g(x)=x-7

Possible Answers:

Correct answer:

Explanation:

First, solve for g(4) .

g(4)=4-7=-3

Next, solve for f(-3) .

f(-3)=(-3)^2=9

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Example Question #164 : Algebra

Define an operation $\Cup$ as follows:

$a \Cup b = 0$ if either or , but not both, are negative, and $a \Cup b = a+ b$ otherwise.

Evaluate:

$\left [(-8) \Cup (-8) \right ]+\left ( 8\Cup8 \right )$

Possible Answers:

Correct answer:

Explanation:

Evaluate each of $(-8) \Cup (-8)$ and $8\Cup8 \right$ separately. Since in both cases, the numbers have the same sign, replace the numbers for and in the second expression - that is, simply add them.

$\left [(-8) \Cup (-8) \right ]+\left ( 8\Cup8 \right )$

$= \left [ (-8 )+ (-8) \right ] + (8+8)$

= -16 + 16 = 0

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Example Question #168 : Algebra

Define an operation $\pounds$ as follows:

For all real numbers a,b ,

$a \pounds b = -ab$ if a > b , and $a \pounds b = ab$ if $a \leq b$

Evaluate:

$\left ( 5 \pounds 3\right )\times \left ( 3 \pounds 5\right )$

Possible Answers:

225

Correct answer:

Explanation:

Evaluate $5 \pounds 3$ and $3 \pounds 5$ separately.

Since 5 > 3 , use

$a \pounds b = -ab$ for $5 \pounds 3$

$5\pounds 3 = - 5 \times 3 = -15$

Since 3 <5 , use

$a \pounds b = ab$ for $3 \pounds 5$

$3 \pounds 5 = 3 \times 5 = 15$

$\left ( 5 \pounds 3\right )\times \left ( 3 \pounds 5\right ) = -15 \times 15 = -225$

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GMAT Math : Algebra

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #161 : Algebra

Example Question #162 : Algebra

Example Question #21 : Understanding Functions

Example Question #21 : Understanding Functions

Example Question #22 : Understanding Functions

Example Question #23 : Understanding Functions

Example Question #163 : Algebra

Example Question #24 : Understanding Functions

Example Question #164 : Algebra

Example Question #168 : Algebra

All GMAT Math Resources

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