GMAT Math : GMAT Quantitative Reasoning

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Example Questions

Example Question #65 : Functions/Series

Define function as follows:

$f(x) = x^{2} - 12x - 17$

has an inverse on each of the following domains except:

Possible Answers:

None of the other choices gives a correct answer.

$x \in (0,5)$

$x \in (-10, -5)$

$x\in (5, 10)$

$x \in (-5,0)$

Correct answer:

$x\in (5, 10)$

Explanation:

has an inverse on a given domain if and only if there are no two distinct values on the domain a,b such that f(a) = f(b) .

$f(x) = x^{2} - 12x - 17$ is a quadratic function, so its graph is a parabola. The key is to find the -coordinate of the vertex of the parabola, which can be found by completing the square:

$f(x) = x^{2} - 12x + \left ( \frac{- 12}{2} \right )^{2} - 17 - \left ( \frac{- 12}{2} \right )^{2}$

$f(x) = x^{2} - 12x +36 - 17 - 36$

$f(x) = \left (x-6 \right )^{2} - 53$

The vertex occurs at x = 6 , so the interval which contains this value will have at least one pair a,b such that f(a) = f(b) . The correct choice is $x\in (5, 10)$ ..

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Example Question #66 : Functions/Series

Define function as follows:

$f(x) = \cos \frac{x}{3}$

has an inverse on each of the following domains except:

Possible Answers:

$x\in (9,12)$

$x \in (3,6)$

$x\in (12,15)$

$x \in (6,9)$

$x\in (0,3)$

Correct answer:

$x\in (9,12)$

Explanation:

has an inverse on a given domain if and only if there are no two distinct values on the domain a,b such that f(a) = f(b) .

$f(x) = \cos \frac{x}{2}$ has a sinusoidal wave as its graph, with period $\frac{2 \pi }{\frac{1}{3}} =6 \pi$ ; it begins at a relative maximum of (0,1) and has a relative maximum or minimum every $3\pi$ units. Therefore, any interval containing an integer multiple of $3\pi$ will have at least two distinct values a,b such that f(a) = f(b) .

The only interval among the choices that includes a multiple of $3 \pi$ is (9,12) :

$3 \pi \approx 3 \cdot 3.14 \approx 9.42 \in (9,12)$ .

This is the correct choice.

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Example Question #67 : Functions/Series

What is the sum of all terms from 0 to 30 inclusive ?

Possible Answers:

350

262

309

465

435

Correct answer:

465

Explanation:

We could solve this by actually adding up all terms the from 0 to 30, but it would take way too much time. There is a simple formula to remember for the summations of consecutive terms: $\frac{n^{2}+n}{2}$ , which gives the sum of all terms from 0 to .

By substituting the value provided in our problem (30) into the formula, we can solve for the correct answer.

$\frac{30^{2}+30}{2}$

$\frac{900+30}{2}$

$\frac{930}{2}$

465

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Example Question #62 : Functions/Series

What is the sum of the even terms from 2 to 60?

Possible Answers:

465

900

600

630

930

Correct answer:

930

Explanation:

We should notice that since we have a sequence of even numbers, we can factor out, so we can rewrite it as :

2+4+6+...+60=2(1+2+3+... + 30)

We can calculate the summation of all numbers from 1 to with the formula $\frac{n^{2}+n}{2}$ ; so, we simply have to plug in 30 for and multiply this formula by two:

$2(\frac{30^{n}+30}{2})= 930$

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Example Question #211 : Algebra

What is the sum of the sequence of all terms from 120 to 160 inclusive?

Possible Answers:

7140

5740

5840

2870

12880

Correct answer:

5740

Explanation:

The formula for the summation of consecutive terms is $\frac{n^{2}+n}{2}$ , which gives the sum of all terms from 0 to . We can apply the formula to get the summation of all consecutive terms from 1 to 160. To figure out the summation starting from 120, we simply have to subtract the summation of all terms from 1 to 119. (We don't want to include 120 since we want it in our summation.)

$\frac{160^{2}+160}{2}-\frac{119^{2}+119}{2} = 12880-7140= 5740$

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Example Question #1291 : Problem Solving Questions

What is the sum of all odd numbers from 0 to 59 inclusive?

Possible Answers:

731

870

900

873

360

Correct answer:

900

Explanation:

We can manipulate summations to make them easier to work with. Here, we are asked for the sum of odd terms from 1 to 59. We can calculate this by subtracting the summation of the even terms from the summation of all numbers from 1 to 59, using the formula $\frac{n^{2}+n}{2}$ to sum all terms from 1 to .

In other words, we have to calculate $\frac{59^{2}+59}{2}-\frac{2\cdot (29^{2}+29)}{2}$ , since the even numbers are given by 2+4+6...+58 or 2(1+2+3...+29) .

We obtain the final answer 900.

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Example Question #212 : Algebra

Define f (x) = x - |x | . Which of the following would be a valid alternative way of expressing the definition of ?

Possible Answers:

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

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

$f (x) = \left\{\begin{matrix} 1 & \textrm{ if } x < 1\\ 2x & \textrm{ if } x \geq 1 \end{matrix}\right.$

$f (x) = \left\{\begin{matrix} 1 & \textrm{ if } x < 1\\ x & \textrm{ if } x \geq 1 \end{matrix}\right.$

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

Correct answer:

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

Explanation:

By definition:

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

If x < 0 , then |x| = -x ,and subsequently, $f (x) = x- \left ( - x \right ) = 2x$

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Example Question #73 : Understanding Functions

Piecewise

Let be the piecewise-defined function graphed above. Define the function $g (x) = x^{2} - 3$ .

Evaluate $\left (g \circ f \right ) (1)$ .

Possible Answers:

Correct answer:

Explanation:

$\left (g \circ f \right ) (1) = g [f(1)]$

As seen in the diagram below, the graph of includes the point (1, -9) .

Piecewise 1

Therefore, f(1) = -9 , and

$\left (g \circ f \right ) (1) = g (-9)$ .

$g (x) = x^{2} - 3$ , so

$g (-9) = (-9)^{2} - 3 = 81 - 3 = 78$ .

Therefore, $\left (g \circ f \right ) (1) = 78$ , the correct choice.

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Example Question #74 : Understanding Functions

Piecewise

Let be the piecewise-defined function graphed above. Define the function $g (x) = x^{2} - 3$ .

Evaluate $\left (f \circ g \right ) (1)$ .

Possible Answers:

Correct answer:

Explanation:

$\left (f \circ g \right ) (1) = f [g(1)]$

$g (x) = x^{2} - 3$ , so

$g (1) = 1^{2} - 3 = 1-3 = -2$ , so

$\left (f \circ g \right ) (1) = f (-2)$

As seen in the diagram below, the graph of includes the point (-2, -1) .

Piecewise 1

Therefore, f(-2) = -1 , and $\left (f \circ g \right ) (1) = -1$ , the correct choice.

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Example Question #75 : Understanding Functions

Piecewise

Let be the piecewise-defined function graphed above. Define function $g(x) = x^{2} - 11$ .

Evaluate $\left (g \circ f^{-1} \right ) (4)$ .

Possible Answers:

$45\frac{1}{2}$

$-10\frac{3}{4}$

4 is not in the domain of $g \circ f^{-1}$

Correct answer:

$-10\frac{3}{4}$

Explanation:

$\left (g \circ f^{-1} \right ) (4) = g [f ^{-1} (4)]$

$f ^{-1} (4) = c$ such that f(c)= 4 .As seen in the diagram below, the graph of includes the point $\left (\frac{1}{2}, 4 \right )$ , so $f ^{-1} (4) = \frac{1}{2}$ .

Piecewise 1

$\left (g \circ f^{-1} \right ) (4) = g \left ( \frac{1}{2} \right )$

$g(x) = x^{2} - 11$ , so

$g\left ( \frac{1}{2} \right )=\left ( \frac{1}{2} \right )^{2} - 11 = \frac{1}{4} -11 = -10 \frac{3}{4}$

$\left (g \circ f^{-1} \right ) (4) =-10 \frac{3}{4}$ , the correct choice.

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GMAT Math : GMAT Quantitative Reasoning

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #65 : Functions/Series

Example Question #66 : Functions/Series

Example Question #67 : Functions/Series

Example Question #62 : Functions/Series

Example Question #211 : Algebra

Example Question #1291 : Problem Solving Questions

Example Question #212 : Algebra

Example Question #73 : Understanding Functions

Example Question #74 : Understanding Functions

Example Question #75 : Understanding Functions

All GMAT Math Resources

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