GMAT Math : Algebra

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Example Questions

Example Question #1261 : Gmat Quantitative Reasoning

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

What is $f ^{-1}$ ?

Possible Answers:

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

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

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

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

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

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 #37 : 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 = 7

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

y=137

y = 49

y=8

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 #181 : Algebra

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 #181 : Algebra

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 #183 : Algebra

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

What is $f ^{-1}$ ?

Possible Answers:

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

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

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

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

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

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 #182 : Algebra

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

What is $f ^{-1}$ ?

Possible Answers:

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

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

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

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

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

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 #183 : Algebra

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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Example Question #1271 : Problem Solving Questions

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

Define $f (x) = \left \lfloor x ^{2} \right \rfloor - 4 x + 7$

Evaluate $f \left ( \frac{7}{5} \right )$

Possible Answers:

$f \left ( \frac{7}{5} \right ) = 3 \frac{2}{5}$

$f \left ( \frac{7}{5} \right ) = 2$

$f \left ( \frac{7}{5} \right ) = 1 \frac{2}{5}$

$f \left ( \frac{7}{5} \right ) = 3$

$f \left ( \frac{7}{5} \right ) = 2 \frac{2}{5}$

Correct answer:

$f \left ( \frac{7}{5} \right ) = 2 \frac{2}{5}$

Explanation:

$f (x) = \left \lfloor x ^{2} \right \rfloor - 4 x + 7$

$f \left ( \frac{7}{5} \right ) = \left \lfloor\left ( \frac{7}{5} \right )^{2} \right \rfloor - 4\cdot \frac{7}{5}+ 7$

$= \left \lfloor \frac{49}{25} \right \rfloor - \frac{28}{5}+ 7$

$= \left \lfloor 1 \frac{24}{25} \right \rfloor -5 \frac{3}{5}+ 7$

$=1 -5 \frac{3}{5}+ 7 = 2 \frac{2}{5}$

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Example Question #182 : Algebra

$f (x) = \left\{\begin{matrix} 2x + 1 & x < 0 \\ 3x - 1 & x \geq 0 \end{matrix}\right.$

$\small g (x) = \left\{\begin{matrix} 3x - 1 & x < 0 \\ 4x +7 & x \geq 0 \end{matrix}\right.$

Evaluate $\small \small (f \circ g) (4) - (g \circ f) (-4)$ .

Possible Answers:

113

162

Correct answer:

Explanation:

$\small \small \small \small (f \circ g) (4) = f (g (4)) = f (4 \cdot 4 + 7) = f (23) = 3 \cdot 23 - 1 = 68$

$\small \small \small \small (g \circ f) (-4) = g (f (-4)) = g (2 \cdot (-4) + 1)$ $\small \small = g (-7) = 3 (-7) -1 = -22$

$\small \small \small \small (f \circ g) (4) - (g \circ f) (-4) = 68 - (-22) = 90$

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Example Question #183 : Algebra

For any real a,b , define $a\odot b =(1-a)(1-b)$ .

For what value or values of would $x\odot x = 4$ ?

Possible Answers:

-1, 3

$-\frac{1}{2}, \frac{1}{2}$

No such value of exists.

$1- \sqrt{2}, 1+ \sqrt{2}$

-2,2

Correct answer:

-1, 3

Explanation:

For such an to exist, it must hold that $x\odot x = (1 -x )(1-x) =(1-x)^{2}=4$ .

Take the square root of both sides:

1-x = -2 or 1-x = 2

Case 1:

1-x = -2

-x = -3

x=3

Case 2:

1-x = 2

-x = 1

x=-1

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GMAT Math : Algebra

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #1261 : Gmat Quantitative Reasoning

Example Question #37 : Functions/Series

Example Question #181 : Algebra

Example Question #181 : Algebra

Example Question #183 : Algebra

Example Question #182 : Algebra

Example Question #183 : Algebra

Example Question #1271 : Problem Solving Questions

Example Question #182 : Algebra

Example Question #183 : Algebra

All GMAT Math Resources

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