GMAT Math : Functions/Series

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Example Question #1 : Understanding Functions

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?

Possible Answers:

$14$

$20$

$15$

$5$

$25$

Correct answer:

$14$

Explanation:

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.

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Example Question #1 : Understanding Functions

There exists a set = {1, 2, 3, 4}. Which of the following defines a function of ?

Possible Answers:

none are functions

$f=\left \{ (2,3),(1,4),(2,1),(3,2),(4,4) \right \}$

two are functions

$h=\left \{ (2,1),(3,4),(1,4),(2,1),(4,4) \right \}$

$g=\left \{ (3,1),(4,2),(1,1) \right \}$

Correct answer:

$h=\left \{ (2,1),(3,4),(1,4),(2,1),(4,4) \right \}$

Explanation:

Let's look at $f,g,and\ h$ and see if any of them are functions.

1. = {(2, 3), (1, 4), (2, 1), (3, 2), (4, 4)}: This cannot be a function of because two of the ordered pairs, (2, 3) and (2, 1) have the same number (2) as the first coordinate.

2. = {(3, 1), (4, 2), (1, 1)}: This cannot be a function of because it contains no ordered pair with first coordinate 2. Because the set = {1, 2, 3, 4}, we need an ordered pair of the form (2, ) .

3. = {(2, 1), (3, 4), (1, 4), (2, 1), (4, 4)}: This is a function. Even though two of the ordered pairs have the same number (2) as the first coordinate, is still a function of because (2, 1) is simply repeated twice, so the two ordered pairs with first coordinate 2 are equal.

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Example Question #1 : Understanding Functions

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

Possible Answers:

$y=x^{2}$

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

all are appropriate ways to define

$x\rightarrow x^{2}$

$f(y)=x^{2}$

Correct answer:

$f(y)=x^{2}$

Explanation:

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, .

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Example Question #1 : Understanding Functions

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

Possible Answers:

2x+h

$x^{2}$

$x^{2}+4x+4$

$x^{2}+2xh+h^{2}$

Correct answer:

2x+h

Explanation:

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

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Example Question #1 : Understanding Functions

Give the range of the function:

$f (x) = \left\{\begin{matrix} 2x - 1 & & x < -2\\ -3x& & -2\leq x\leq 2\ \\5 & & \; x > 2 \end{matrix}\right.$

Possible Answers:

$(-\infty , 6]$

$[-6, \infty )$

$(-\infty , 5]$

$(-\infty , -6] \cup [-5, 6]$

$[-5, \infty)$

Correct answer:

$(-\infty , 6]$

Explanation:

We look at the range of the function on each of the three parts of the domain. The overall range is the union of these three intervals.

On $(-\infty , -2)$ , f (x) takes the values:

x < -2

2x - 1 < 2 (-2) - 1

2x - 1 < -5

$f\left (x \right )< -5$

or $(-\infty , -5)$

On [-2, 2] , f (x) takes the values:

$-2 \leq x\leq 2$

$-3(-2) \geq -3x\geq -3 (2)$

$6 \geq -3x\geq -6$

$-6 \leq f(x)\leq 6$ ,

or [-6, 6]

On $(2, \infty)$ , f (x) takes only value 5.

The range of f (x) is therefore $(-\infty , -5) \cup [-6, 6] \cup \left \{ 5\right \}$ , which simplifies to $(-\infty , 6]$ .

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Example Question #1 : Understanding Functions

A sequence begins as follows:

8, 12, ...

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

Possible Answers:

18, 27

16, 20

8, 4

20, 32

20, 35

Correct answer:

20, 32

Explanation:

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

8 + 12= 20

12+20=32

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Example Question #7 : Understanding Functions

Give the inverse of f (x) = 5x - 7

Possible Answers:

$f^{-1}(x)= -\frac{1}{5}x +7$

$f^{-1}(x)=- \frac{1}{5}x + \frac{7}{5}$

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

$f^{-1}(x)= \frac{1}{5}x +7$

$f^{-1}(x)= \frac{1}{5}x + \frac{7}{5}$

Correct answer:

$f^{-1}(x)= \frac{1}{5}x + \frac{7}{5}$

Explanation:

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

y = 5x - 7

x = 5y - 7

x + 7 = 5y - 7 + 7

x + 7 = 5y

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

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

$f^{-1}(x)= \frac{1}{5}x + \frac{7}{5}$

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Example Question #8 : Understanding Functions

Define $f (x) = x^{3} - 8$ . Give $f^{-1} (x)$

Possible Answers:

$f^{-1}(x) = \sqrt[3]{x +8}$

$f^{-1}(x) = 8+ \sqrt[3]{x }$

$f^{-1}(x) = 2- \sqrt[3]{x }$

$f^{-1}(x) = 2+ \sqrt[3]{x }$

$f^{-1}(x) = \sqrt[3]{x +2}$

Correct answer:

$f^{-1}(x) = \sqrt[3]{x +8}$

Explanation:

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

$y = x^{3} - 8$

$x= y^{3} - 8$

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

$x +8= y^{3}$

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

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

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Example Question #1 : Understanding Functions

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

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

Possible Answers:

$(f \circ g)(x) = x^{4}+8x^{2}-12$

$(f \circ g)(x) = x^{4} -16$

$(f \circ g)(x) = x^{4}+8x^{2}+12$

$(f \circ g)(x) = x^{4} - 8x^{2} +20$

$(f \circ g)(x) = x^{4}$

Correct answer:

$(f \circ g)(x) = x^{4}+8x^{2}+12$

Explanation:

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

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

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

$=x^{4} + 8x^{2}+16-4$

$=x^{4} + 8x^{2}+12$

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Example Question #10 : Understanding Functions

Define g (x) = 5x - 2 .

If g (A) = 11 , evaluate .

Possible Answers:

A=2.6

A=57

A=1.8

A=4.2

A=53

Correct answer:

A=2.6

Explanation:

Solve for in this equation:

g (A) = 5A - 2 = 11

5A - 2 + 2 = 11+ 2

5A = 13

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

A = 2.6

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GMAT Math : Functions/Series

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #1 : Understanding Functions

Example Question #1 : Understanding Functions

Example Question #1 : Understanding Functions

Example Question #1 : Understanding Functions

Example Question #1 : Understanding Functions

Example Question #1 : Understanding Functions

Example Question #7 : Understanding Functions

Example Question #8 : Understanding Functions

Example Question #1 : Understanding Functions

Example Question #10 : Understanding Functions

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