Given the following functions f(x) and g(x), solve f[g(6)].
f(x) = 6x + 12
g(x) = x - 8

Answer:

a- p(x) = 2xy(2x -y)(2x + y)(4x² + y²) ⇒ 2nd answer

b- repeated differences of squares ⇒ 1st answer

Step-by-step explanation:

* Lets explain how to solve the problem

∵ p(x) = 32x^5y - 2xy^5

- The coefficients of the two terms are 32 and 2

∵ 2 is a common factor in 32 and 2

∵ 32 ÷ 2 = 16 and 2 ÷ 2 = 1

∴ p(x) = 2(16x^5y - xy^5)

* Now lets find the common factors of the variables x and y

∵ The common factor is x^5 and x is x

∵ The common factor in y and y^5 is y

∴ the common factors in 16x^5y - xy^5 are xy

∵ 16x^5y ÷ xy = 16x^4

∵ xy^5 ÷ xy = y^4

∴ p(x) = 2xy(16x^4 - y^4)

- Remember that a² - b² is called difference of two squares we

 factorize it by distributed into two polynomials have same terms

 with different middle sign (a + b)(a - b)

∵ 16x^4 - y^4 is a different of two squares because √(16x^4) = 4x²

  and √9y^4) = y²

∴ The factorization of 16x^4 - y^4 is (4x² + y²)(4x² - y²)

∴ p(x) = 2xy[(4x² + y²)(4x² - y²)]

- The bracket 4x² - y² is also different of two squares because

 √(4x²) = 2x and √(y²) = y

∴ The factorization of 4x² - y² is (2x - y)(2x + y)

∴ p(x) = 2xy(2x -y)(2x + y)(4x² + y²)

a- p(x) = 2xy(2x -y)(2x + y)(4x² + y²)

b- The methods used to factor p(x) are:

    greatest common factor and repeated differences of squares

- But you ask to chose one answer of b so chose repeated

 differences of squares

Answer:

[tex]\frac{2\pi}{3}[/tex]

Step-by-step explanation:

So here they are asking for one angle that gives you the cosine value is equal to -1/2 while being in quadrant 2.

So you look in quadrant 2 and find x=-1/2 which is at [tex]\frac{2\pi}{3}[/tex].

[tex]\cos^{-1}(\frac{-1}{2})=\frac{2\pi}{3}[/tex] while being in quadrant 2.

Answer:

(4x -1  ) ( x  +5  )

Step-by-step explanation:

4x2 + 19x-5

(4x   ) ( x    )

We need to get -5    The factors are -1 and 5  and 1 and -5

4*5 = 20  

so the 5 will be on the outside  We do this because we have +19 on the inside

(4x -1  ) ( x  +5  )

Lets check

FOIL

4x^2 +20x -x -5 = 4x^2 +19x-5  

Answer:

The factors are (4x-1)(x+5).

Step-by-step explanation:

4x^2+19x-5

Break the middle term:

First multiply the coefficient of first term by the constant.

4*5=20

Now find any two numbers whose product = 20 and whose sum is equal to the middle term.

20*1 = 20

20-1 = 19

Now break the middle term:

4x^2+20x-x-5

Now group the first two terms and last two terms:

(4x^2+20x)-(x+5)

Now take the common from each group.

4x(x+5)-1(x+5)

(4x-1)(x+5).

Thus the factors are (4x-1)(x+5).....

Answer:

The system is inconsistent because there is no solution.  

Step-by-step explanation:

I'm going to put both of these in slope-intercept form (y=mx+b where m is slope and b is y-intercept).

3x-6y=20

Solve for y.

3x-6y=20

Subtract 3x on both sides:

  -6y=-3x+20

Divide both sides by -6:

    y=(-3/-6)x+(20/-6)

Reduce:

  y=(1/2)x+(-10/3)

The slope is 1/2 and the y-intercept is -10/3.

2x-4y=3

Solve for y.

2x-4y=3

Subtract 2x on both sides:

   -4y=-2x+3

Divide both sides by -4:

    y=(-2/-4)x+(3/-4)

Reduce:

   y=(1/2)x+(-3/4)

The slope is 1/2 and the y-intercept is -3/4.

The lines are parallel so they have no intersection. I know they are parallel because they have the same slope and different y-intercept.

The system is inconsistent because there is no solution.  

Answer:

B) Inconsistent

Step-by-step explanation:

Step 1: Write both equations

3x - 6y = 20

2x - 4y = 3

Step 2: Find x in terms of y

3x - 6y = 20

x = 20+6y/3

Step 3: Substitute x in one of the equations to find y

2x - 4y = 3

2(20+6y/3) - 5y = 3

40 + 12y - 12y = 9

0 ≠ -31

Therefore, these system of equations have no solution.

Step 4: Choose the option

A) consistent- A consistent system of equations has at least one set of equations. Therefore, this option is incorrect.

B) inconsistent- An inconsistent system of equations has no solution. Therefore, this option is correct.

C) Independent- An independent system of equations has one solution. Therefore, this option is incorrect.

D) Dependent- A dependent system of equations have infinite solutions. Therefore, this option is incorrect.

Option B is the right answer

!!

Answer:

128

Step-by-step explanation:

You find the number of subsets of a set by using the formula [tex]2^{\text{ number of elements}[/tex].

We have 7 elements so that means we have [tex]2^{7}=128[/tex] subsets.

A set with 7 elements can have 128 subsets, as calculated using the equation 2^n, where n is the number of elements in the set.

In Mathematics, a subset is a portion of a set, including the empty set and the set itself. The number of subsets a set can have is calculated using the equation 2^n, where 'n' is the number of elements in the set. In this case, the set A includes 7 elements, therefore applying the equation we get: 2^7=128 subsets, which is the answer to your question.

For example, {-3, -1, 0} would be a subset of set A, as would {0, 1, 2}, {-2, 0, 2}, and so on. The complete set ({-3,-2,-1,0,1,2,3}) is also a subset, as is the empty set ({}).

https://brainly.com/question/31739353

#SPJ2

Answer:

The first one

Step-by-step explanation:

Using the tangent ratio in the right triangle

tan30° = [tex]\frac{opposite}{adjacent}[/tex] = [tex]\frac{AC}{BC}[/tex] = [tex]\frac{b}{8}[/tex]

Multiply both sides by 8

8 × tan30° = b, that is

b = 8tan30°

Answer:

b = (8)tan(30o)

Step-by-step explanation:

For this case we have that the equation of a line of the slope-intersection form is given by:[tex]y = mx + b[/tex]

Where:

m: It's the slope

b: It is the cut-off point with the y axis

To find the slope we look for two points through which the line passes:

We have to:

[tex](x1, y1) :( 1,2)\\(x2, y2): (- 1, -4)[/tex]

Thus, the slope is:

[tex]m = \frac {-4-2} {- 1-1} = \frac {-6} {- 2} = 3[/tex]

We have then:

[tex]y = 3x + b[/tex]

Substituting a point in the equation to find b:

[tex]2 = 3 (1) + b\\2 = 3 + b\\b = 2-3\\b = -1[/tex]

Finally, the equation is:[tex]y = 3x-1[/tex]

Answer:

Option D

f(x) = 2x + 7 and f(x) = -1

Replace f(x) with -1:

-1 = 2x +7

Subtract 7 from both sides:

-8 = 2x

Divide both sides by 2:

x = -8 /2

x = -4

Answer:

The answer is C) 13.

Step-by-step explanation:

To find the hypotenuse (longest side) of a right triangle, you need to use the Pythagorean Theorem. The formula for the hypotenuse is a^2 + b^2 = c^2.

5^2 (five squared) equals 25, and 12^2 (twelve squared) equals 144.

25 + 144 = 169

Next, find the square root of 169. It is 13. 13 is the length of the hypotenuse.

I hope this helped! :)

In the right triangle ABC, the length of the AC is equal to [tex]13[/tex] units.

" Right triangle is defined as a triangle with one of the interior angles with measure equals to [tex]90[/tex] degrees."

Formula used

Pythagoras theorem,

(Hypotenuse)² = ( adjacent side)² + (opposite side)²

According to the question,

In the right triangle ABC,

Adjacent side 'BC' [tex]= 12[/tex] units

Opposite side 'AB' [tex]= 5[/tex] units

'AC' represents the hypotenuse of the right triangle

Substitute the value in the Pythagoras theorem we get,

[tex]AC^{2} = BC^{2} +AB^{2} \\\\\implies AC^{2} = 12^{2} + 5^{2} \\\\\implies AC^{2} = 144 + 25\\\\\implies AC = \sqrt{169} \\\\\implies AC =13units[/tex]

Hence, Option(C) is the correct answer.

Learn more about the right triangle here

https://brainly.com/question/6322314

#SPJ2

Answer: It’s D

5x

25x can still be simplified all the way down to 5x

(D) 5x Is the correct answer

Answer:

B. 3

Step-by-step explanation:

A common difference in an arithmetic sequence is the value the terms in the sequence vary from one to another.You can find one term to another by adding or subtracting the common difference.

In this case, the terms are 2,5,8,11,.....

From the first term to the second term, the difference is, 5-2=3

From the second term to the third term, the difference is, 8-5=3

From the third term to the fourth term the difference is=11-8=3

Hence the common difference in this arithmetic sequence is 3

[tex]\huge{\text{Hey there!}}[/tex]

[tex]\huge\text{We could use the word {difference} in this equation}[/tex]

[tex]\huge\text{The\ difference usually means subtract in math}}[/tex] [tex]\huge\text{terms!}[/tex]

[tex]\huge{\text{So, we subtract the terms to find your answer!}}[/tex]

[tex]\huge\text{5 - 2 = 3. So we know that 3 could be a possible}[/tex] [tex]\huge\text{answer.}[/tex]

[tex]\huge\text{11 - 8 = 3 }[/tex]

[tex]\huge\rightarrow\text{8 - 5 = 3}[/tex]

[tex]\huge\text{The ratio difference = 3!}[/tex]

[tex]\boxed{\boxed{\huge{\text{Answer: B. 3}}}}\huge\checkmark[/tex]

[tex]\text{Good luck on your assignment and enjoy your day!}[/tex]

~[tex]\frak{LoveYourselfFirst:}[/tex]

Answer:

a)The equation is $35l-$215

b)Amount = $695

c)They will loose $215.

Step-by-step explanation:

From the question you should understand that;

The equation that relates the fund raised and number of lawns aerated during a day is;

Let number of lawns aerated that day be l

The cost to aerate one lawn per day is $35

Cost to rent the aeration machine is $215

a)The equation is ;

[tex]=(l*35)-215[/tex]

=  $35l-$215

b)Money raised

number of lawns,l=26

Substitute in the equation

Amount=$35l-$215

=$(35×26)-$215

=$910-$215=$695

c)If the team were unable to aerate due to weather condition, they will incur a loss to to renting of the aeration machine.They will lose $215.This can be avoided by not renting the aeration machine when the weather conditions seems unfavorable for working.

Answer: Each kid would receive 13 pieces of candy

Step-by-step explanation: Since Carlene has 104 pieces of candy and she wants to distribute them evenly to 8 kids she would have to do division and divide 104 with 8 which equals to 13 which means 13 pieces of candy are going to each kid.

Answer: Our required equation would be

[tex]\dfrac{\text{Number of pieces of candy}}{\text{Number of kids}}[/tex]

and each kid would get 13 pieces of candy.

Step-by-step explanation:

Since we have given that

Number of pieces of candy left over from halloween = 104

Number of kids in which she would like to distribute = 8

As every student get pieces of candy evenly.

So, Number of pieces of candy each kid would get is given by

[tex]\dfrac{\text{Number of pieces of candy}}{\text{Number of kids}}\\\\=\dfrac{104}{8}\\\\=13[/tex]

Hence, our required equation would be

[tex]\dfrac{\text{Number of pieces of candy}}{\text{Number of kids}}[/tex]

and each kid would get 13 pieces of candy.

Answer: 7

Step-by-step explanation: Since there are no 0’s, just count the number of digits. There are 7.

Answer:

25

Step-by-step explanation:

Since she predicts that 80% of the invited people come, she needs to invite a number greater than 20, so that 80% of that number is 20.

Let the number of invited people be x.

80% of x must be 20.

80% * x = 20

0.8x = 20

Divide both sides by 0.8

x = 25

Answer: She should invite 25 people.

Check: Let's find 20% of 25. If it is 20, then 25 is correct.

80% of 25 = 0.8 * 25 = 20

Since 80% of 25 is 20, the answer, 25, is correct.

Answer:

80/100=.80

.80x=20

x=25

Step-by-step explanation:

Answer:

60

Step-by-step explanation:

Answer:

$10.

Step-by-step explanation:

Let x represent the original cost of bag of cat food.

We have been given that a pet store owner set the price of a bag of cat food at 50% above the cost. So the cost of cat food would be x plus 50% of x.

[tex]x+\frac{50}{100}*x\rightarrow x+0.5x=1.5x[/tex]

We are further told that the price was reduced by 20%, when it did not sell. So price of cat food after reduction would be [tex]1.5x[/tex] minus 20 percent of [tex]1.5x[/tex].

[tex]1.5x-(\frac{20}{100}\times 1.5x)[/tex]

[tex]1.5x-(0.20\times 1.5x)[/tex]

[tex]1.5x-(0.3x)[/tex]

[tex]1.2x[/tex]

Since price after reduction was $12, so we will equate [tex]1.2x[/tex] with 12 as:

[tex]1.2x=12[/tex]

[tex]\frac{1.2x}{1.2}=\frac{12}{1.2}[/tex]

[tex]x=10[/tex]

Therefore, the original cost of cat food was $10.

Answer:

Option D is correct

Step-by-step explanation:

The original polynomial is:

–2x2 + 2 + 5x3 – 5x

Arranging in decreasing power of x:

[tex]5x^3 - 2x^2 -5x+2[/tex]

Factoring the given polynomial by grouping:

[tex]5x^3 - 2x^2 -5x+2\\=5x^3-5x- 2x^2+2\\=5x(x^2-1)-2(x^2-1)\\=(5x-2)(x^2-1)[/tex]

Now, (x^2-1) can be further solved using formula:

(a^2-b^2)=(a-b)(a+b)

Solving:

[tex]=(5x-2)(x^2-1)\\=(5x-2)(x-1)(x+1)[/tex]

So, [tex](5x-2)(x-1)(x+1)[/tex] represents the factors of [tex]-2x2 + 2 + 5x3-5x[/tex]

Hence Option D is correct.

Answer: the answer is D

Step-by-step explanation:

the screen has proof it is correct... also please STOP deleting my answers its all ways the same person and its getting really annoying when im giving you the correct answer, im just trying to help :(

Answer:

Possibility 1: [tex]\frac{(x+1)(x+1)}{(x-4)(x+5)}[/tex]

Possibility 2: [tex]\frac{(x+1)(x+3)(x-3)}{x(x-4)(x+5)}[/tex]

Possibility 3: [tex]\frac{-4(x-4)(x+1)}{-4(x-4)(x-4)}[/tex]

There are infinitely many more possibilities.

Step-by-step explanation:

So we are looking for a fraction in terms of x.

We have a horizontal asymptote so that means the degree of the top has to equal to degree of the bottom. It is also at y=1 which means the coefficient of the leading term on top and bottom must be the same (but not zero) since the same number divided by the same number is 1.  

Now we also have a vertical asymptote at x=4 which means we need a factor of x-4 on bottom.

So there is a lot of possibilities. Here are a few:

Possibility 1: [tex]\frac{(x+1)(x+1)}{(x-4)(x+5)}[/tex]

You have the degrees are the same on top and bottom and the leading coefficients are the same.  You also have that factor of (x-4) on bottom.

Possibility 2: [tex]\frac{(x+1)(x+3)(x-3)}{x(x-4)(x+5)}[/tex]

Possibility 3: [tex]\frac{-4(x-4)(x+1)}{-4(x-4)(x-4)}[/tex]

Now a factor of (x-4) can be canceled here but you still have a factor of (x-4) left on bottom so you still have the vertical asymptote. You also still have the same leading coefficient on top and bottom (-4 in this case) and the same degree.

Answer:

12.53

Step-by-step explanation:

x = x₂ - x₁ = -6 - 0   = -6

y = y₂ - y₁ =  7 - (-4) = 11

We can use the Pythagorean theorem to calculate the distance.

z² = (-6)² + 11² = 36 + 121 = 157

z = √157 ≈ 12.53

The distance between the two points is 12.53.

Answer:

[tex]\sqrt[3]{x} -3[/tex]: The parent graph will move 3 units down.

[tex]\sqrt[3]{x-3}[/tex]: The parent graph will move 3 units right.

Step-by-step explanation:

The parent function [tex]\sqrt[3]{x}[/tex] looks like the first image attached.

Using knowledge of transformations, numbers outside the radical move the graph up or down (vertically). A positive number moves the graph up; a negative number moves the graph down.

Numbers inside the radical move the graph left or right (horizontally). A positive number moves the graph left; a negative number moves the graph right.

In the first transformation [tex]\sqrt[3]{x} -3[/tex] the number is outside of the radical, meaning that the graph will move vertically. The 3 is negative, so the graph will move 3 units down.

The first transformation moves the parent graph 3 units down.

In the second transformation [tex]\sqrt[3]{x-3}[/tex] the number is inside the radical so the graph will move horizontally. The 3 is negative so the graph will move 3 units to the right.

The second transformation moves the parent graph 3 units right.

The attached images show:

Answer:

7033/10000

Step-by-step explanation:

You get his answer by times it by 1000 so the decimal point is out of the number

Answer:

The domain but not the range of the transformed function is the same as that of the parent function ⇒ answer D

Step-by-step explanation:

* Lets talk about the transformation

- If the function f(x) reflected across the x-axis, then the new

 function g(x) = - f(x)

- If the function f(x) translated horizontally to the right  by h units,

 then the new function g(x) = f(x - h)

- The domain of a function is set of the values of x which make

  the function defined

- The range is the set values of y that corresponding with the domain

- The domain of the function f(x) = IxI is the set of all real numbers

∴ The domain of f(x) is {x : x ∈ R}

- The range of  the function f(x) = IxI is the set of all real numbers

  greater than or equal 0

∴ The range f(x) = {y : y ≥ 0}

∵ f(x) reflected across the x-axis, then it will be change to g(x) = -IxI

∴ All the y-coordinates of the point on the function will be change

  from positive values to negative values

∵ The rang of f(x) is {y : y ≥ 0}

∴ The range of g(x) is {y : y ≤ 0}

∵ After the reflection the function translated 6 units to the right

∴ The x will change to x - 6

∴ The function will be h(x) = -Ix - 6I

- There is no values of x make h(x) undefined, then its domain is

  set of all real number

∴ The domain of h(x) is {x : x ∈ R}

∵ The domain of f(x) is {x : x ∈ R}

∵ The range of h(x) is the same the range of g(x)

∴ The range of h(x) is {y : y ≤ 0}

- f(x) and h(x) have same domains and different ranges

∴ The correct statement is: The domain but not the range of the

   transformed function is the same as that of the parent function

- Look to the attached graph for more understanding

# The red graph is f(x)

# The blue graph is h(x)

Answer:

d

Step-by-step explanation:

Answer:

[tex]h= 116.616[/tex]

Step-by-step explanation:

we have

[tex]h= 105sin (0.0698(90+1)) + 105[/tex]

step 1

Solve (90+1)

[tex]h= 105sin (0.0698(91)) + 105[/tex]

step 2

Solve 0.0698(91)

[tex]h= 105sin (6.3518) + 105[/tex]

step 3

Solve sin (6.3518)

[tex]h= 105(0.11063) + 105[/tex]

step 4

Solve 105(0.11063)

[tex]h= 11.616 + 105[/tex]

step 5

Solve 11.616 + 105

[tex]h= 116.616[/tex]

Answer:

x = 40

Step-by-step explanation:

The sum of the 3 angles in a triangle = 180°

Consider the triangle on the left and calculate the third angle

Subtract the sum of the 2 angles from 180

third angle = 180° - (75 + 50)° = 180° - 125° = 55°

Consider the triangle on the right

The third angle = 55° ( vertical angles )

Subtract the sum of the 2 angles from 180

x = 180 - (55 + 85) = 180 - 140 = 40

Check the picture below.

bear in mind that, horizonta lines are just y = "some constant", and vertical lines are x = "some constant".

Answer:

AB = 3

Step-by-step explanation:

Since AB is a perpendicular bisector then Δ is isosceles and AD = AC, that is

4x - 1 = 2x + 3 ( subtract 2x from both sides )

2x - 1 = 3 ( add 1 to both sides )

2x = 4 ( divide both sides by 2 )

x = 2

Hence AB = x + 1 = 2 + 1 = 3

Answer:

AB=3

Step-by-step explanation:

Answer:

True

Step-by-step explanation:

The prefix "co" means together, and the base "linear" means line. Therefore co-linear means together on a line

Answer:

(g-f)(3) = 23

Step-by-step explanation:

f(x) = 4-x^2

g(x) = 6x

we need to find g-f(3)

Put the value of x = 3 in g(x) and f(x) and subtract f(x) from g(x) i.e

(g-f)(3) = g(3) - f(3)

(g-f)(3) = 6(3) - {4-(3)^2}

(g-f) (3)=18-(4-9)

(g-f)(3) = 18-(-5)

(g-f)(3) = 18+5

(g-f)(3) = 23

so, (g-f)(3) = 23

Answer:

23

Step-by-step explanation:

(g-f)(3)=g(3)-f(3)

So we need to find g(3) and f(3). Then subtract.

g(3)=6(3)=18.

f(3)=4-(3)^2=4-9=-5.

(g-f)(3)=g(3)-f(3)=18-(-5)=18+5=23

Answer:

The ordered pair of P'=(-8.75, 5.25)

Step-by-step explanation:

To get the ordered pair of point P' you simply have to multiply the coordinates of P by scale factor of 7/4

P:(-5,3)

P'(x,y)= (-5* 7/4, 3*7/4)

P'(x,y) = (-35/4 , 21/4)

P'(x,y) = (-8.75, 5.25)..

Thus the ordered pair of P'=(-8.75, 5.25)....

Answer:

(−5.25, −3.5)

Step-by-step explanation:

Answer:

E and F.

Step-by-step explanation:

Let the side opposite the smallest angle have a measurement of x (this is the shortest side).

The hypotenuse is twice the shorter side so the hypotenuses would be 2x in this case.

The long leg is square root of 3 times the short side or sqrt(3)x in this case.

So the ratio of long leg to short leg is [tex]\frac{\sqrt{3}x}{x}=sqrt(3):1[/tex]

and

the ratio of short leg to long leg is [tex]\frac{x}{\sqrt{3}x}=1:sqrt(3)[/tex].

The answer is not A that division is equivalent to 1:1.

B same reason as A.

C is not right because 1:sqrt(2) is not the same as 1:sqrt(3)

I bolded the reason in C so you can see why I said that.

You can also put these in your calculator and compare the decimals like so:

D gives us 0.816 approximately while  sqrt(3)/1 gives 1.73 and 1/sqrt(3) gives 0.58 approximately. 0.816 is neither one of those.

How about E?  1:sqrt(3) is exactly what one of our ratios say.

How about F?  sqrt(3)/3=0.58 so this is what one of our ratios is equivalent to.

So E and F are your answers.

The only correct options are: E. [tex]\(1 : \sqrt{3}\)[/tex] and F. [tex]\(\sqrt{3} : 3\)[/tex].

To determine the correct ratios between the lengths of the legs of a [tex]\(30^\circ\)-\(60^\circ\)-\(90^\circ\)[/tex] triangle, we first recall the properties of such a triangle:

Step 1: In a [tex]\(30^\circ\)-\(60^\circ\)-\(90^\circ\)[/tex] triangle, the side lengths are in the ratio [tex]\(1:\sqrt{3}:2\)[/tex], where:

The shortest leg (opposite the [tex]\(30^\circ\)[/tex] angle) has length (1).

The longer leg (opposite the [tex]\(60^\circ\)[/tex] angle) has length [tex]\(\sqrt{3}\)[/tex].

The hypotenuse has length (2).

Step 2: Identify the correct ratio between the legs:

The ratio of the shortest leg to the longer leg is [tex]\(1:\sqrt{3}\)[/tex].

Now, let's analyze the given options:

A. [tex]\(\sqrt{2} : \sqrt{2}\)[/tex]: This ratio simplifies to (1:1), which is incorrect for a [tex]\(30^\circ\)-\(60^\circ\)-\(90^\circ\)[/tex] triangle.

B. [tex]\(\sqrt{3} : \sqrt{3}\)[/tex]: This ratio simplifies to (1:1), which is incorrect for a [tex](30^\circ\)-\(60^\circ\)-\(90^\circ\)[/tex] triangle.

C. [tex]\(1 : \sqrt{2}\)[/tex]: This does not match the ratio [tex]\(1 : \sqrt{3}\)[/tex].

D. [tex]\(\sqrt{2} : \sqrt{3}\)[/tex]: This does not match the ratio [tex]\(1 : \sqrt{3}\)[/tex].

E. [tex]\(1 : \sqrt{3}\)[/tex]: This matches the correct ratio [tex]\(1 : \sqrt{3}\)[/tex].

F. [tex]\(\sqrt{3} : 3\)[/tex]: This does not match the ratio [tex]\(1 : \sqrt{3}\)[/tex].

Thus, the only correct options are: E. [tex]\(1 : \sqrt{3}\)[/tex] and F. [tex]\(\sqrt{3} : 3\)[/tex]