GMAT Math : Problem-Solving Questions

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Example Questions

Example Question #36 : Functions/Series

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

For all real numbers a,b ,

$a \triangleright b = \frac{a}{b}$ if and only if $b \neq 0$ , and $a \triangleright b = a ^{2}$ otherwise.

Evaluate:

$\left (5 \triangleright 2 \right )-\left ( 2 \triangleright 5 \right )$

Possible Answers:

- 2.1

2.1

Correct answer:

2.1

Explanation:

We can evaluate both $5 \triangleright 2 \right$ and $2 \triangleright 5$ using the first part of the definition, since the second number is nonzero in both cases.

$a \triangleright b = \frac{a}{b}$

$5\triangleright 2 = \frac{5}{2} = 2.5$

$2\triangleright 5 = \frac{2}{5} = 0.4$

$\left (5 \triangleright 2 \right )-\left ( 2 \triangleright 5 \right ) = 2.5 - 0.4 = 2.1$

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Example Question #37 : Functions/Series

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

$a \bullet b = a ^{2} (b- 10)$ for all real a,b .

Evaluate:

$10 \bullet 5$

Possible Answers:

500

Correct answer:

Explanation:

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

$10 \bullet 5 = 10 ^{2} (5- 10) = 100 \cdot (-5) = -500$

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Example Question #38 : Functions/Series

Define f (x) = 5x + 9 .

What is $f ^{-1}$ ?

Possible Answers:

$f ^{-1} (x)=- \frac{1}{5}x + \frac{9}{5}$

$f ^{-1} (x)= \frac{1}{5}x - \frac{9}{5}$

$f ^{-1} (x)=-5x -9$

$f ^{-1} (x)=- \frac{1}{5}x - \frac{9}{5}$

$f ^{-1} (x)= \frac{1}{5}x - 9$

Correct answer:

$f ^{-1} (x)= \frac{1}{5}x - \frac{9}{5}$

Explanation:

Replace f(x) with .

y = 5x + 9

Now exchange and and solve for in the new expression:

x = 5y + 9

x - 9 = 5y + 9 - 9

x - 9 = 5y

$\left (x - 9 \right ) \div 5 = 5y \div 5$

$\frac{1}{5}x - \frac{9}{5} = y$

$f ^{-1} (x)= \frac{1}{5}x - \frac{9}{5}$

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Example Question #39 : Functions/Series

Define $f (x) = x ^{3}$

What is $f ^{-1}$ ?

Possible Answers:

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

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

$f (x) = - x ^{3}$

$f ^{-1} (x) = \sqrt[3]{x}$

$f ^{-1} (x) = - \sqrt[3]{x}$

Correct answer:

$f ^{-1} (x) = \sqrt[3]{x}$

Explanation:

Replace f(x) with .

$y= x ^{3}$

Now exchange and and solve for in the new expression:

$x= y^{3}$

$\sqrt[3]{x}= \sqrt[3]{y^{3}}$

$\sqrt[3]{x}=y$

$f^{-1}(x)= \sqrt[3]{x}$

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Example Question #40 : Functions/Series

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

$a \bigstar b = 100-ab$ for all real numbers a,b

Solve for :

$y \bigstar 7 = 44$

Possible Answers:

y=8

y = 49

y = 7

$y = 10 \frac{4}{7}$

y=137

Correct answer:

y=8

Explanation:

Substitute a=y,b=7 into the definition and solve:

$a \bigstar b = 100-ab$

$y \bigstar 7 = 44$

$100- y \cdot 7 = 44$

100- 7 y = 44

100- 7 y -100 = 44-100

- 7 y = -56

$- 7 y\div (-7) = -56 \div (-7)$

y=8

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Example Question #41 : Understanding Functions

Define $f(x)= x ^{3} - 125$ .

What is $f ^{-1}$ ?

Possible Answers:

$f ^{-1} (x)= \sqrt[3]{x}+125$

$f ^{-1} (x)= \sqrt[3]{x+5}$

$f ^{-1} (x)=\frac{1}{x ^{3} - 125}$

$f ^{-1} (x)= \sqrt[3]{x+125}$

$f ^{-1} (x)= \sqrt[3]{x}+5$

Correct answer:

$f ^{-1} (x)= \sqrt[3]{x+125}$

Explanation:

Replace f(x) with .

$y= x ^{3} - 125$

Now exchange and and solve for in the new expression:

$x= y^{3} - 125$

$x + 125= y^{3} - 125 + 125$

$x + 125= y^{3}$

$\sqrt[3]{x + 125 }= y$

$f ^{-1} (x)= \sqrt[3]{x+125}$

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Example Question #42 : Understanding Functions

Define an operation $\oplus$ on the set of positive integers as follows:

$a \oplus b = ab$ if and are both odd or both even; otherwise, $a \oplus b = 0$ .

Evaluate:

$\left ( 5 \oplus 5\right ) + \left ( 6 \oplus 6\right )$

Possible Answers:

6 1

Correct answer:

6 1

Explanation:

Both $5 \oplus 5$ and $6 \oplus 6$ can be defined using the first part of the definition of $a \oplus b$ , as in the first case, both numbers are odd, and in the second case, both are even.

$a \oplus b = ab$

$5 \oplus 5 = 5 \times 5 = 25$

$6 \oplus 6 =6 \times 6 = 36$

Add:

$\left ( 5 \oplus 5\right ) + \left ( 6 \oplus 6\right ) = 25 + 36 = 61$

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Example Question #43 : Understanding Functions

Define $f (x) = \frac{5}{4-x}$ .

What is $f ^{-1}$ ?

Possible Answers:

$f^{-1}(x)=\frac{5-x }{4}$

$f^{-1}(x)=\frac{ 5x -4 }{x }$

$f^{-1}(x)=\frac{ 4x -5 }{x }$

$f^{-1}(x)=\frac{ 4-x }{5}$

$f^{-1}(x)=\frac{ 4x -1 }{5x }$

Correct answer:

$f^{-1}(x)=\frac{ 4x -5 }{x }$

Explanation:

Replace f(x) with .

$y = \frac{5}{4-x}$

Now exchange and , and solve for in the new expression:

$x = \frac{5}{4-y}$

$x \left ( 4-y \right ) = \frac{5}{4-y} \left ( 4-y \right )$

$x \left ( 4-y \right ) = 5$

$\frac{x \left ( 4-y \right ) }{x }=\frac{ 5 }{x }$

$4-y=\frac{ 5 }{x }$

$4-y +y -\frac{ 5 }{x } =\frac{ 5 }{x }+y -\frac{ 5 }{x }$

$4-\frac{ 5 }{x } =y$

$f^{-1}(x)= 4-\frac{ 5 }{x } =\frac{ 4x }{x } -\frac{ 5 }{x } = \frac{ 4x -5 }{x }$

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Example Question #44 : Understanding Functions

Define $f (x) = \frac{1}{7} x- 13$ .

What is $f ^{-1}$ ?

Possible Answers:

$f ^{-1} (x) = 7 x + 91$

$f ^{-1} (x) = -\frac{1}{7} x + 13$

$f ^{-1} (x) = - 7 x + 91$

$f ^{-1} (x) = -\frac{1}{7} x + 91$

$f ^{-1} (x) = 7 x + 13$

Correct answer:

$f ^{-1} (x) = 7 x + 91$

Explanation:

Replace f(x) with .

$y = \frac{1}{7} x- 13$

Now exchange and and solve for in the new expression:

$x= \frac{1}{7} y- 13$

$x + 13 = \frac{1}{7} y- 13 + 13$

$x + 13 = \frac{1}{7} y$

$\left ( x + 13 \right ) \cdot 7 = \frac{1}{7} y \cdot 7$

7 x + 91 = y

Therefore,

$f ^{-1} (x) = 7 x + 91$ .

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Example Question #45 : Understanding Functions

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

For all real a,b ,

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

Evaluate:

$49 \bigstar (-64)$

Possible Answers:

Correct answer:

Explanation:

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

$49 \bigstar (-64) = 49 + \sqrt{\left | -64 \right |} = 49 + \sqrt{64 } = 49 + 8 = 57$

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All GMAT Math Resources

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GMAT Math : Problem-Solving Questions

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #36 : Functions/Series

Example Question #37 : Functions/Series

Example Question #38 : Functions/Series

Example Question #39 : Functions/Series

Example Question #40 : Functions/Series

Example Question #41 : Understanding Functions

Example Question #42 : Understanding Functions

Example Question #43 : Understanding Functions

Example Question #44 : Understanding Functions

Example Question #45 : Understanding Functions

All GMAT Math Resources

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