Understanding functions - GMAT Quantitative

Card 0 of 688

There is water tank already $\frac{4}{7}$ full. If Jose adds 5 gallons of water to the water tank, the tank will be $\frac{13}{14}$ full. How many gallons of water would the water tank hold if it were full?

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In this case, we need to solve for the volume of the water tank, so we set the full volume of the water tank as x. According to the question, $\frac{4}{7}$-full can be replaced as $\frac{4}{7}$x. $\frac{13}{14}$-full would be $\frac{13}{14}$x. Therefore, we can write out the equation as:

$\frac{4}{7}$x+5=\frac{13}{14}$x.

Then we can solve the equation and find the answer is 14 gallons.

Let be a function that assigns $x^{2}$ to each real number . Which of the following is NOT an appropriate way to define ?

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This is a definition question. The only choice that does not equal the others is $f(y)=x^{2}$. This describes a function that assigns $x^{2}$ to some number , instead of assigning $x^{2}$ to its own square root, .

If $f(x)=x^{2}$, find $\frac{f(x+h)-f(x)}{h}$.

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We are given f(x) and h, so the only missing piece is f(x + h).

$f(x+h)=(x+h)^{2}$$=x^{2}$$+2xh+h^{2}$

Then $\frac{f(x+h)-f(x)}{h}$= $$\frac{x^{2}$$$+2xh+h^{2}$$-x^{2}$}{h} = $$\frac{2xh+h^{2}$$}{h}=2x+h

A sequence begins as follows:

It is formed the same way that the Fibonacci sequence is formed. What are the next two numbers in the sequence?

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Each term of the Fibonacci sequence is formed by adding the previous two terms. Therefore, do the same to form this sequence:

For any values , , define the operation $\Gamma$ as follows:

$a; \Gamma; b = 2ab - a - b$

Which of the following expressions is equal to $2x; \Gamma; 2y$ ?

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Substitute and for and in the expression for $a; \Gamma; b$ :

$2x; \Gamma; 2y = 2(2x)(2y) - 2x - 2y = 8xy -2x-2y$

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

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

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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:

or

Case 1:

Case 2:

Define $f(x) = $x^{2}$- 4$ and $g(x) = $x^{2}$ + 4$ .

Give the definition of $(f \circ g)(x)$ .

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$(f \circ g)(x) = f(g(x))$

$= $f(x^{2}$+4)$

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

Give the inverse of

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The easiest way to find the inverse of is to replace in the definition with , switch with , and solve for in the new equation.

$$\frac{1}{5}$ \cdot\left ( x + 7 \right ) = $\frac{1}{5}$ \cdot 5y$

$$\frac{1}{5}$ x + $\frac{7}{5}$ = y$

Define $f (x) = $x^{3}$ - 8$ . Give

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The easiest way to find the inverse of is to replace in the definition with , switch with , and solve for in the new equation.

$y = $x^{3}$ - 8$

$x= $y^{3}$ - 8$

$x +8= $y^{3}$ - 8 +8$

$x +8= $y^{3}$$

$\sqrt[3]{x +8}= $\sqrt[3]{y^{3}$}$

$y = \sqrt[3]{x +8}$

Define an operation $\odot$ as follows:

For any real numbers ,

$a \odot b = (a-1) (b-1)$

Evaluate $\left (5 \odot 5 \right )\odot 5$ .

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$\left (5 \odot 5 \right )\odot 5$

$= \left [(5 -1) (5 -1) \right ] \odot 5$

$= (4 \cdot 4 ) \odot 5$

$= 16 \odot 5$

$= \left (16-1 \right ) \left (5-1 \right )$

$= 15 \cdot4 = 60$

Define $f(x) = A $(x-B)^{3}$$ , where $A \neq 0, B \neq 0$ .

Evaluate in terms of and .

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This is equivalent to asking for the value of for which , so we solve for in the following equation:

$A $(x-B)^{3}$ =1$

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

$x-B =\sqrt[3]{$\frac{1}{A}$}$

$x-B = $\frac{1}{\sqrt[3]{A}$}$

$x-B + B = B + $\frac{1}{\sqrt[3]{A}$}$

$x = B + $\frac{1}{\sqrt[3]{A}$}$

Therefore, .

A sequence is formed the same way the Fibonacci sequence is formed. Its third and fourth terms are 16 and 30, respectively. What is its first term?

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A Fibonacci-style sequence starts with two numbers, with each successive number being the sum of the previous two. The second term is therefore the difference of the fourth and third terms, and the first term is the difference of the third and second.

Second term:

First term:

An infinite sequence begins as follows:

Assuming this pattern continues infinitely, what is the sum of the 1000th, 1001st and 1002nd terms?

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This can be seen as a sequence in which the term is equal to if is not divisible by 3, and otherwise. Since 1,000 and 1,001 are not multiples of 3, but 1,002 is, the 1000th, 1001st, and 1002nd terms are, respectively,

and their sum is

$1000+ 1001+ \left (-1002 \right ) = 999$

Define an operation $\ddagger$ on the set of real numbers as follows:

$a \ddagger b = 2ab - a - b$

Evaluate $4 \ddagger (3\ddagger 2 )$ .

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First, evaluate $3\ddagger 2$ by substituting :

$a \ddagger b = 2ab - a - b$

$3 \ddagger 2 = 2\cdot 3 \cdot 2 - 3 - 2 = 12 - 3 - 2 = 7$

Second, evaluate $4\ddagger 7$ in the same way.

$a \ddagger b = 2ab - a - b$

$4\ddagger 7 = 2 \cdot 4 \cdot 7 - 4 - 7 = 56 - 4 - 7 = 45$

Define .

If , evaluate .

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Solve for in this equation:

$5A \div 5 = 13\div 5$

Define the operation $\Theta$ as follows:

$a ; \Theta ; b = ab + 100$

Solve for : $4; \Theta ; x = 72$

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$4; \Theta ; x = 72$

$4x \div 4 = -28 \div 4$

Define an operation $\diamond$ as follows:

For any real numbers ,

$A \diamond B = $A^{2}$$+B^{2}$$ .

Evaluate $\left (5 \diamond 0 \right ) + \left (-5 \diamond 0 \right )$ .

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$5 \diamond 0 = $5^{2}$ - 0 = 25$

$-5 \diamond 0 = \left (-5 \right $)^{2}$ - 0 = 25$

$\left (5 \diamond 0 \right ) + \left (-5 \diamond 0 \right ) = 25 + 25 = 50$

Define $f(x) = $\sqrt{x}$$ . What is $\left (f \circ f \right )(x)$ ?

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This can best be solved by rewriting $\sqrt{ x}$ as and using the power of a power property.

$\left (f \circ f \right )(x) = f (f(x)) = \left [f(x) $\right]^{$\frac{1}${2}$} =\left ( $x^{$\frac{1}${2}$} \right $)^{$\frac{1}$${2}$}=x^{$\frac{1}${2}$\cdot $\frac{1}{2}$ } $=x^{$\frac{1}${4}$ } =\sqrt[4]{x}$

$\left \lceil x \right \rceil$ is defined as the least integer greater than or equal to .

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

Define $g (x) = \left \lfloor x+ 0.5 \right \rfloor - \left \lceil x - 0.5 \right \rceil$ .

Evaluate .

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$g (x) = \left \lfloor x+ 0.5 \right \rfloor - \left \lceil x - 0.5 \right \rceil$

$g (3.9) = \left \lfloor 3.9+ 0.5 \right \rfloor - \left \lceil 3.9 - 0.5 \right \rceil$

$g (3.9) = \left \lfloor 4.4 \right \rfloor - \left \lceil 3.4 \right \rceil$

$\left \lceil x \right \rceil$ is defined as the least integer greater than or equal to .

Define $f(x) = \left \lceil x ^{2}\right \rceil$ .

Define $g(x) = \left \lceil x + 0.5\right \rceil - 0.5$ .

Evaluate $\left ( f \circ g \right )(4)$ .

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$\left ( f \circ g \right )(4) = f(g(4))$

First, evaluate :

$g(x) = \left \lceil x + 0.5\right \rceil - 0.5$

$g(4) = \left \lceil 4 + 0.5 \right \rceil - 0.5$

$g(4) = \left \lceil 4.5 \right \rceil - 0.5$

Now, evaluate :

$f(x) = \left \lceil x ^{2}\right \rceil$

$f(4.5) = \left \lceil 4.5 ^{2}\right \rceil$

$f(4.5) = \left \lceil 20.25 \right \rceil = 21$