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Identify the domain of the following rational function
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25 Apr 2021, 11:34

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Identify the domain of the following rational function

\(f(x)=\frac{x^3-9x}{\sqrt{3x^2-6x-9}}\)

Select all that apply

A. -1

B. 0

C. 1

D. 2

E. 3

F. 5

\(f(x)=\frac{x^3-9x}{\sqrt{3x^2-6x-9}}\)

Select all that apply

A. -1

B. 0

C. 1

D. 2

E. 3

F. 5

ShowHide Answer

Official Answer

B,C,D,F

Re: Identify the domain of the following rational function
[#permalink]
22 Feb 2024, 00:46

1

Note that \(f(x)\) is a rational function. Rational functions are characterized by the ratio of two polynomials, and in this case, \(f(x)\) is defined as:

\[ f(x) = \frac{x^3 - 9x}{\sqrt{3x^2 - 6x - 9 \]

Although the presence of the square root in the denominator introduces a slight deviation from the most traditional form of a rational function (typically expressed as a simple ratio \(p(x)/q(x)\) without square roots), the function can still be considered within the broader family of rational functions due to its nature of involving a ratio of polynomial expressions, with the caveat that the denominator includes a radical expression. This necessitates additional considerations for the domain, specifically ensuring that the expression under the square root is positive to keep the function real and defined.

### Summary and Clarification

- **The function is indeed a rational function** with additional constraints due to the square root in the denominator.

- **The domain of \(f(x)\) requires** that the denominator is both real and non-zero. This is satisfied for \(x > 3\) and \(x < -1\).

- **At \(x = 0\),** despite the numerator being \(0\), the function is undefined because the denominator involves a square root of a negative number, resulting in an imaginary value, which is not permissible in the real function domain.

- **Therefore, only option F. 5** is valid for the domain of \(f(x)\), satisfying the condition for the function to be real and defined.

This clarification reinforces the original conclusion regarding the domain and the valid option based on the function's characteristics.

\[ f(x) = \frac{x^3 - 9x}{\sqrt{3x^2 - 6x - 9 \]

Although the presence of the square root in the denominator introduces a slight deviation from the most traditional form of a rational function (typically expressed as a simple ratio \(p(x)/q(x)\) without square roots), the function can still be considered within the broader family of rational functions due to its nature of involving a ratio of polynomial expressions, with the caveat that the denominator includes a radical expression. This necessitates additional considerations for the domain, specifically ensuring that the expression under the square root is positive to keep the function real and defined.

### Summary and Clarification

- **The function is indeed a rational function** with additional constraints due to the square root in the denominator.

- **The domain of \(f(x)\) requires** that the denominator is both real and non-zero. This is satisfied for \(x > 3\) and \(x < -1\).

- **At \(x = 0\),** despite the numerator being \(0\), the function is undefined because the denominator involves a square root of a negative number, resulting in an imaginary value, which is not permissible in the real function domain.

- **Therefore, only option F. 5** is valid for the domain of \(f(x)\), satisfying the condition for the function to be real and defined.

This clarification reinforces the original conclusion regarding the domain and the valid option based on the function's characteristics.

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Identify the domain of the following rational function
[#permalink]
26 Apr 2021, 05:47

3

Carcass wrote:

Identify the domain of the following rational function

\(f(x)=\frac{x^3-9x}{\sqrt{3x^2-6x-9}}\)

Select all that apply

A. -1

B. 0

C. 1

D. 2

E. 3

F. 5

\(f(x)=\frac{x^3-9x}{\sqrt{3x^2-6x-9}}\)

Select all that apply

A. -1

B. 0

C. 1

D. 2

E. 3

F. 5

Domain is defined as all real values of \(x\) (input) for which the function does not fail. The function can fail in two cases (i) either \(f(x)\) is imaginary or (ii) \(f(x)\) is not defined for some values of \(x\)

\(f(x)=\frac{x^3-9x}{\sqrt{3x^2-6x-9}}\)

Here, \(\sqrt{3x^2-6x-9}\) cannot be equal to zero, otherwise the function would be not defined

\(\sqrt{3(x^2-2x-3)} ≠ 0\)

\(\sqrt{3(x^2-3x-x-3)} ≠ 0\)

\(\sqrt{3(x-3)(x+1)} ≠ 0\)

i.e. \(x ≠ 3\) or \(x ≠ -1\)

Hence, option B, C, D, and F

Identify the domain of the following rational function
[#permalink]
Updated on: 25 May 2021, 02:02

Carcass

Does having sqaure root of denominator count in rational function? Like x=1

Does having sqaure root of denominator count in rational function? Like x=1

Re: Identify the domain of the following rational function
[#permalink]
24 May 2021, 20:09

2

Shouldn't what in the square root be greater than 0, and so the answer should be F only?

Re: Identify the domain of the following rational function
[#permalink]
25 May 2021, 02:20

Expert Reply

Darsh12 wrote:

Carcass

Does having sqaure root of denominator count in rational function? Like x=1

Does having sqaure root of denominator count in rational function? Like x=1

yes sir

see our math book for a deep understanding about roots

https://gre.myprepclub.com/forum/gre-math- ... -2609.html

ask if you need more

Re: Identify the domain of the following rational function
[#permalink]
25 May 2021, 02:21

Expert Reply

tranhaianh1405 wrote:

Shouldn't what in the square root be greater than 0, and so the answer should be F only?

not quite sure what you meant Sir

Re: Identify the domain of the following rational function
[#permalink]
25 May 2021, 09:45

Carcass wrote:

tranhaianh1405 wrote:

Shouldn't what in the square root be greater than 0, and so the answer should be F only?

not quite sure what you meant Sir

I mean for example suppose that B is correct, then plugging in the denominator would be the square root of -9 and that does not make sense because under the root it should be a positive value.

Identify the domain of the following rational function
[#permalink]
25 May 2021, 10:55

Expert Reply

tranhaianh1405 wrote:

Carcass wrote:

tranhaianh1405 wrote:

Shouldn't what in the square root be greater than 0, and so the answer should be F only?

not quite sure what you meant Sir

I mean for example suppose that B is correct, then plugging in the denominator would be the square root of -9 and that does not make sense because under the root it should be a positive value.

Good question

So

\(\sqrt{-9}\)

\(\sqrt{(9 \times -1)}\)

\(\sqrt{9} \times \sqrt{-1}\)

\(3 \times \sqrt{-1}\) and \(\sqrt{-1}=i\)

so in the end

\(\sqrt{-9}\)

is basically \(3i\)

So we would have \(\frac{0}{3i}\) = correct option because is 0

Re: Identify the domain of the following rational function
[#permalink]
22 Jun 2021, 17:07

3

tranhaianh1405 wrote:

Carcass wrote:

tranhaianh1405 wrote:

Shouldn't what in the square root be greater than 0, and so the answer should be F only?

not quite sure what you meant Sir

I mean for example suppose that B is correct, then plugging in the denominator would be the square root of -9 and that does not make sense because under the root it should be a positive value.

Good question

So

\(\sqrt{-9}\)

\(\sqrt{(9 \times -1)}\)

\(\sqrt{9} \times \sqrt{-1}\)

\(3 \times \sqrt{-1}\) and \(\sqrt{-1}=i\)

so in the end

\(\sqrt{-9}\)

is basically \(3i\)

So we would have \(\frac{0}{3i}\) = correct option because is 0[/quote]

I agree with what tranhaianh1405 says about the domain of this rational expression; the ETS "Graduate Record Examinations Mathematical Conventions" (google the title...I do not have enough of a post history to include links) state that we are to assume that all numbers used in the test questions are real numbers, and that when the square root symbol is included in an expression, it means the "non negative square root with the domain >= 0:

"1. All numbers used in the test questions are real numbers. In particular, integers and both rational and irrational numbers are to be considered, but imaginary numbers are not. This is the main assumption regarding numbers. Also, all quantities are real numbers, although quantities may involve units of measurement. "

"5. Here are nine examples of other standard symbols with their meanings:

...

Example 5: [sqrt(x)] the nonnegative square root of x, where [x>=0]"

"6. Because all numbers are assumed to be real, some expressions are not defined. Here are three examples:

...

Example 2: If x<0 then [sqrt(x)] is not defined."

"9. Standard function notation is used in the test, as shown in the following three examples.

Example 1: The function g is defined for all [x >= 0] by [g(x) = 2x + sqrt(2)]

Example 2: If the domain of a function f is not given explicitly, it is assumed to be the set of all real numbers x for which f(x) is a real number. "

----------------------------------------

With this in mind, it seems that if we are assuming that the given expression is a "real" function, that the domain would be all real x<-1 and x>3.

This means that the only given answer item that falls into the function's domain is "E".

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Re: Identify the domain of the following rational function
[#permalink]
03 Jul 2021, 04:02

1

Theory: Domain of a function f(x) is the set of all possible values of x for which f(x) has a real value

\(f(x)=\frac{x^3-9x}{\sqrt{3x^2-6x-9}}\)

Now, f(x) is a fraction and has a square root.

So, f(x) will not be real if the denominator is 0 or the expression inside the square root evaluates to < 0

=> All values of x for which \(3x^2-6x-9 <=0 \) will not be in Domain.

\(3x^2-6x-9 <=0 \)

Divide both the sides by 3 we get

\(x^2-2x-3 <=0 \)

=> \(x^2 + x -3x -3 <=0 \)

=> x(x+1) -3(x+1) <= 0

=> (x+1) * (x-3) <= 0 [ To learn how to Solve Inequalities watch this video ]

=> -1 <= x <= 3

But for x=0 we will get numerator 0, making the expression 0 anyways.

So, Domain of f(x) = All real values of x except -1 <= x < 0 and 0 < x <= 3

So, Answer will be B and F I think.

Hope it helps!

To learn more about Functions and Inequalities watch the following video

\(f(x)=\frac{x^3-9x}{\sqrt{3x^2-6x-9}}\)

Now, f(x) is a fraction and has a square root.

So, f(x) will not be real if the denominator is 0 or the expression inside the square root evaluates to < 0

=> All values of x for which \(3x^2-6x-9 <=0 \) will not be in Domain.

\(3x^2-6x-9 <=0 \)

Divide both the sides by 3 we get

\(x^2-2x-3 <=0 \)

=> \(x^2 + x -3x -3 <=0 \)

=> x(x+1) -3(x+1) <= 0

=> (x+1) * (x-3) <= 0 [ To learn how to Solve Inequalities watch this video ]

=> -1 <= x <= 3

But for x=0 we will get numerator 0, making the expression 0 anyways.

So, Domain of f(x) = All real values of x except -1 <= x < 0 and 0 < x <= 3

So, Answer will be B and F I think.

Hope it helps!

To learn more about Functions and Inequalities watch the following video

Re: Identify the domain of the following rational function
[#permalink]
03 Jul 2021, 06:53

how can option b can give a valid answer.

it gives denominator as square root of -9 which is. undefined

it gives denominator as square root of -9 which is. undefined

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Joined: **02 Jan 2020 **

Status:**GRE Quant Tutor**

Posts: **1093**

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GMAT 1: **700 Q51 V31**

GPA: **2.8**

WE:**Engineering (Computer Software)**

Re: Identify the domain of the following rational function
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03 Jul 2021, 18:50

1

aliceeee : This is because \(\frac{0}{i}\) same as 0*i is equal to zero.

gmatclubot

Re: Identify the domain of the following rational function [#permalink]

03 Jul 2021, 18:50
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