Mathematics College

Find the value of x

Answers

Answer 1

[tex] \qquad \qquad \bf \huge\star \: \: \large{ \underline{Answer} } \huge \: \: \star[/tex]

x = 35°

[tex]\textsf{ \underline{\underline{Steps to solve the problem} }:}[/tex]

[tex]\qquad❖ \: \sf \:4x + 1 + x + 4 = 180[/tex]

[ by linear pair ]

[tex]\qquad❖ \: \sf \:5x + 5 = 180[/tex]

[tex]\qquad❖ \: \sf \:5(x + 1) = 180[/tex]

[tex]\qquad❖ \: \sf \:x + 1 = 180 \div 5[/tex]

[tex]\qquad❖ \: \sf \:x + 1 = 36[/tex]

[tex]\qquad❖ \: \sf \:x = 36 - 1[/tex]

[tex]\qquad❖ \: \sf \:x = 35[/tex]

[tex] \qquad \large \sf {Conclusion} : [/tex]

x = 35°

Related Questions

List these fractions from
least to greatest 3/8 5/8 4/8 2/8 7/8

Answers

Answer: 2/8, 3/8, 4/8, 5/8, 7/8

Step-by-step explanation:

Since they all have a common denominator, we can list them based on their numerator without having to worry about the denominator.

2/8 < 3/8 < 4/8 < 5/8 < 7/8

NO LINKS!! Please help me with this problem

Answers

Answer:

y²/324 -x²/36 = 1

Step-by-step explanation:

Where (0, ±b) are the ends of the transverse axis and y = ±(b/a)x describes the asymptotes, the equation of the hyperbola can be written as ...

y²/b² -x²/a² = 1

Application

Here, we have transverse axis endpoints of (0, ±18) and asymptotes of y = ±3x, so we can conclude ...

b = 18

b/a = 3 ⇒ a = 18/3 = 6

The equation of the hyperbola in standard form is ...

y²/324 -x²/36 = 1

QUESTION 3 The probability that it will rain on a given day is 63%. A child has a 12% chance of falling in dry weather and is three times as likely to fall in wet weather. Draw a tree diagram to represent all outcomes of the above information. What is the probability that a child will not fall on any given day? What is the probability that a child will fall in dry weather? 3.1 draw a tree diagram to represent all outcomes of the above information

Answers

Answer + Step-by-step explanation:

the probability that a child will not fall on any given day :

= (63 × 64 + 27 × 88) ÷ 100

= 64.08 %

the probability that a child will fall in dry weather :

= (27 × 12) ÷ 100

= 3.24 %

A community college employs 88 full-time faculty members. To gain the faculty's opinions about an upcoming building project, the college president wishes to obtain a simple random sample that will consist of 9 faculty members. He numbers the faculty from 1 to 88. Complete parts (a)

-Using the provided random number table, the president closes his eyes and drops his ink pen on the table. It points to the digit in row 3, column 6. Using this position as the starting point and proceeding downward, determine the numbers for the 9 faculty members who will be included in the sample.

Answers

The numbers that would be included in the sample are 4, 1, 1, 4, 7, 9, 2, 6, 4, 7, 4, 4, 1, 0, 3, 0, 5, 0, 7, 1, 1, 6, 5, 4, 9

How to determine the number that will be included in the sample?

The complete question is added as an attachment

The given parameters are:

Row = 3

Column = 6

From the attached figure, the entries in row 3 are:

4, 1, 1, 4, 7, 9, 2, 6, 4, 7, 4, 4, 1, 0, 3, 0, 5, 0, 7 1

From the attached figure, the entries in column 6 are:

1, 6, 9, 5, 4, 9

Hence, the numbers that would be included in the sample are 4, 1, 1, 4, 7, 9, 2, 6, 4, 7, 4, 4, 1, 0, 3, 0, 5, 0, 7, 1, 1, 6, 5, 4, 9

Answers

Answer: 20 and 14

Step-by-step explanation:

Using a little guess and check, we can figure out that the numbers are 6 apart and the difference between their squares is 204.

[tex]20^2 = 400\\14^2=196[/tex]

[tex]20^2-14^2 = 204[/tex]

A trick to doing this is to start with a number that is easily squared like 10 or 20 and see if the difference between their numbers is larger or smaller and change your numbers based on them.

Hope this helped :)

Answer:

14 and 20

Step-by-step explanation:

Let x and y be two numbers where x < y.

We are given :

y - x = 6

y² - x² = 204

Then

y - x = 6

(y - x)(y + x) = 204

Then

y - x = 6

6 × (y + x) = 204

then

y - x = 6 (Equation 1)

y + x = 34 (Equation 2)

Then

(Equation 1) + (Equation 2) ⇒ 2y = 40 ⇒ y = 20

Now ,we substitute y by 20 in equation 1 :

y - x = 6

⇔ 20 - x = 6

⇔ 20 - 6 = x

⇔ x = 14

What is the largest prime number p such that 8 times p is less than 1000?

Answers

Answer:

113

Step-by-step explanation:

first we need to divide 1000 by 8 to get 125.

Our prime number has to be less than 125.

The largest prime number less than 125 is 113.

PLEASE HELP ALOT OF POINYSAshlee is working with a local construction firm to improve Interstate 35.
The construction firm has created steel supports that will be arranged
into triangular shapes with side lengths of 6 feet, 9 feet, and where the
third side of each triangle has various lengths. They have asked Ashlee to
design a package that she could guarantee would fit all the materials. She
decided to use constructions to determine the maximum value for the
unknown side, x, and then ship all her materials in a box of length x feet.
Which is the best conclusion for Ashlee to make?

Answers

The best conclusion for Ashlee to make is (c) Ashlee determined that the box must be less than 15 feet by verifying the triangle inequality theorem.

How to determine the maximum lengths?

The side lengths of the triangle are given as:

6 feet and 9 feet

Let the third side be x.

By triangle inequality theorem, we have:

a + b > c

So, we have

6 + 9 > x

6 + x > 9

9 + x > 6

Evaluate the inequalities

15 > x

x > 3

x > -3

Remove the negative inequality

15 > x

x > 3

Combine the inequalities

3 < x < 15

So, the maximum side is less than 15

Hence, the best conclusion for Ashlee to make is (c) Ashlee determined that the box must be less than 15 feet by verifying the triangle inequality theorem.

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$2,870 per month. He pays $574 a month for rent. What percent of his monthly pay goes to rent?

Answers

we know that 2870 is the 100%, what is 574 off of it in percentage?

[tex]\begin{array}{ccll} amount&\%\\ \cline{1-2} 2870 & 100\\ 574& x \end{array} \implies \cfrac{2870}{574}~~=~~\cfrac{100}{x} \\\\\\ 5=\cfrac{100}{x}\implies 5x=100\implies x=\cfrac{100}{5}\implies x=20[/tex]

The percent of his monthly pay which goes to rent if he earns a total of $2,870 per month is 20%.

What percent of his monthly pay goes to rent?

Amount earned per month = $2,870

Amount paid for rent = $574

Let

The unknown percent = x

x% of $2,870 = $574

x/100 × 2,870 = 574

2,870x/100 = 574

cross product

2,870x = 57,400

divide both sides by 2,870

x = 57,400/2,870

x = 20%

Ultimately, 20% of the monthly pay goes to rent.

Answers

Answer:

Q1: Third Choice. 15√3

Q2: Fourth Choice. 15√3

Q3: First Choice. 12√10

Step-by-step explanation:

Question 1: Which of the following is the solution to 8√3 + 7√3?

Given expression

8√3 + 7√3

Simplify by addition

Remember that terms with the same radical part can add together directly

= (8 + 7) √3

[tex]\Large\boxed{=15\sqrt{3} }[/tex]

Question 2: Which of the following is the solution to √15 * √45 simplified?

Given expression

√15 × √45

Factorize each radical

= √(3 × 5) × √(9 × 5)

= √3 × √5 × √9 × √5

Simplify necessary radicals

= √3 × √5 × 3 × √5

= 3 × √3 × √5 × √5

Simplify by multiplication

When radicals multiply, combine them under one radical sign

= 3√3 × √(5 × 5)

= 3√3 × 5

[tex]\Large\boxed{=15\sqrt{3} }[/tex]

Questions 3: Which of the following is the solution to 15√10 - 3√10?

Given expression

15√10 - 3√10

Simplify by subtraction

Remember that terms with the same radical part can subtract each other directly

= (15 - 3) √10

[tex]\Large\boxed{=12\sqrt{10} }[/tex]

Hope this helps!! :)

Please let me know if you have any questions

Find equations of the following. y = x^2 − z^2, (6, 32, 2) (a) the tangent plane (b) the normal line

Answers

The equation for the tangent plane is 12x - y + 4z - 56 = 0 and the equation for normal line is (x - 6)/12 = (32 - y) = (z - 2)/4

Finding the Equations for Tangent Plane and Normal Line:

The given function is,

f(x, y, z) = x² - y - z² = 0

∂f/ ∂x = 2x

∂f/ ∂y = -1

∂f/ ∂z = 2z

At given point (6, 32, 2),

∂f/ ∂x = 12

∂f/ ∂y = -1

∂f/ ∂z = 4

(a) The equation of tangent plane is given as follows,

(∂f/ ∂x)(x-x₁) + (∂f/ ∂y)(y-y₁) + (∂f/ ∂z)(z-z₁) = 0

12(x - 6) - 1(y - 32) + 4(z - 4) = 0

12x - 72 - y + 32 + 4z - 16 = 0

The required tangent plane is,

12x - y + 4z - 56 = 0

(b) The equation for normal line is given as,

(x-x₁) / (∂f/ ∂x) = (y-y₁) / (y-y₁) = (z-z₁) / (∂f/ ∂z)

(x - 6)/12 = (y - 32)/(-1) = (z - 2)/4

Thus, the required equation of normal line is,

(x - 6)/12 = (32 - y) = (z - 2)/4

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Two regular 6-sided dice are tossed. Compute the probability that the sum of the pops on the upward faces of the 2 dice is the following. 1

Answers

Answer: 0

Step-by-step explanation:

ASAP help me with this ty!

Answers

Answer:

96 degrees

Step-by-step explanation:

The angle bisectors splits the two angles mentioned down the middle so the 2 angles are equal to each other.

4x + 4 = 2(x + 13) Distribute the 2

4x + 4 = 2x + 26 Subtract 2x from both sides

2x + 4 = 26 Subtract 4 from both sides

2x = 22 Divide both sides by 2

x = 11 Plug that back into either the right side or the left side of the original equation

4x + 4

4(11) + 4

44 + 4

48. Each angle is 48 degrees. 48 + 48 is 96

need this answered asap Which function has a minimum and is transformed to the right and down from the parent function, f(x) = x2? g(x) = –9(x + 1)2 – 7 g(x) = 4(x – 3)2 + 1 g(x) = –3(x – 4)2 – 6 g(x) = 8(x – 3)2 – 5

Answers

The function that has a minimum and is transformed to the right and down from the parent function is; g(x) = 8(x² - 3²) - 5

How to Interpret Functions?

By inspection, only Functions B) and D) have a minimum.

For the first function in option B, we have:

g(x) = 4(x²- 6x + 9) + 1

This can be simplified to;

g(x) = 4( x - 3 )² + 1

Thus, we can say that this first function is transformed to the right and up from the parent function.

For the second function in option D, we have;

g(x) = 8(x² - 6x + 9 ) - 5

The function can be simplified to get;

g(x) = 8( x - 3 )² - 5

Thus, we can say that it is transformed to the right and down.

The function that has a minimum and is transformed to the right and down from the parent function is D

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Answer: option D

Step-by-step explanation:

PLEASE HELP ME WITH THIS QUESTION ASAPP

Answers

Answer:

Trapezoid

Step-by-step explanation:

One pair of parallel sides.

Amy and Richard each solved an equation using the quadratic formula.

Amy's Equation and Method

Latex: 4x^2+7x-20=0
4
x
2
+
7
x
−
20
=
0

Step 1: Latex: x=\frac{-7\pm \sqrt{7^2-4(1)(20)}}{2(4)}
x
=
−
7
±
√
7
2
−
4
(
1
)
(
20
)
2
(
4
)

Step 2: Latex: x=\frac{-7\pm \sqrt{49-80}}{8}
x
=
−
7
±
√
49
−
80
8

Step 3: Latex: x=\frac{-7\pm \sqrt{-31}}{8}
x
=
−
7
±
√
−
31
8

Step 4: Latex: x=\frac{-7\pm i\sqrt{31}}{8}
x
=
−
7
±
i
√
31
8

Richard's Equation and Method

Latex: x^2-6x+8=0
x
2
−
6
x
+
8
=
0

Step 1: Latex: x=\frac{6\pm \sqrt{(-6)^2-4(1)(8)}}{2(1)}
x
=
6
±
√
(
−
6
)
2
−
4
(
1
)
(
8
)
2
(
1
)

Step 2: Latex: x=\frac{6\pm \sqrt{36-32}}{2}
x
=
6
±
√
36
−
32
2

Step 3: Latex: x=\frac{6\pm \sqrt{4}}{2}
x
=
6
±
√
4
2

Step 4: Latex: x=\frac{6+\sqrt{4}}{2}
x
=
6
+
√
4
2
and Latex: x=\frac{6-\sqrt{4}}{2}
x
=
6
−
√
4
2

Step 5: Latex: x=\frac{6+2}{2}
x
=
6
+
2
2
and Latex: x=\frac{6-2i}{2}
x
=
6
−
2
i
2

Step 6: Latex: x=4
x
=
4
and Latex: x=3-i
x
=
3
−
i

Both students made a mistake.

Describe the mistake each student made.

Explain what each student needs to do to fix their mistake.

Create your own quadratic equation, and explain how to use the quadratic formula to solve it. Be specific, using Latex: a\textsf{, }b\textsf{,} and Latex: c
c
of your equation and giving the solutions to the equation you chose.

Number your responses from 1 to 3 so your instructor can tell which question you're responding to. You may not receive full credit if your teacher cannot determine that you've answered each question.

Search entries or author

Answers

The mistake each student made and what each student needs to do to fix their mistake is;

Amy didn't include the negative sign for the 20, so she should include it

Richard added a negative sign underneath the radical

Quadratic equation

x² - 6x + 8 = 0

x = - b ± √b² - 4ac / 2a

where,

a = 1b = -6c = 8

x = - b ± √b² - 4ac / 2a

= -(-6) ± √(-6)² - 4(1)(8) / 2(1)

= 6 ± √36 - 32 / 2

= 6 ± √4 / 2

= 6 ± 2 / 2

x = 6/2 + 2/2 or 6/2 - 2/2

= 3 + 1 or 3 - 1

x = 4 or 2

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PLease see attached. This is an algebra question

Answers

The solution for the given expression is 16.

Power Rules

There are different power rules, see some them:

1. Multiplication with the same base: you should repeat the base and add the exponents.

2. Division with the same base: you should repeat the base and subctract the exponents.

3.Power. For this rule, you should repeat the base and multiply the exponents.

4. Zero Exponent. When you have an exponent equals to zero, the result must be 1.

First, you apply the Power Rules - Power for [tex](\frac{2^2x^2y}{xy^3} )^2}[/tex]. For this rule, you should repeat the base and multiply the exponents. Thus, the result will be:[tex]\frac{16x^4y^2}{x^2y^6}[/tex].

After that, you should apply the Power Rules - Division . For this rule, you should repeat the base and subctract the exponents. Thus, the result will be:[tex]\frac{16x^2}{y^4}[/tex].

Now, you should replace the variable x by 4 and the variable y by 2. Thus, the result will be:[tex]\frac{16*4^2}{2^4}=\frac{16*16}{16} =16*1=16[/tex]

Answers

Algebraic expressions are mathematical statements that consist of both number(s) and alphabet(x). Thus the number of stamps in the book is 66[tex]\frac{43}{60}[/tex] (67).

Algebraic expressions are mathematical statements that consist of both number(s) and alphabet(x). Majorly, the alphabet in the expression is referred to as the unknown value which has to be determined.

When the power of any of the alphabets contained in an algebraic expression is more than one, then it can be referred to as a polynomial.

In the given question, let the number of stamps in the book be represented by N.

So that;

N - 5/12 + 24 + 15 + 3/10 + 27 = 0

N - (5/12 + 24 + 15 + 3/10 + 27) = 0

N - [tex]\frac{8006}{120}[/tex] = 0

N - 66[tex]\frac{43}{60}[/tex] = 0

N = 66[tex]\frac{43}{60}[/tex]

= 66.72

N ≅ 67

Therefore the number of stamps in the book is approximately 67.

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sin^4x rewrite the power as a product of two squared terms

Answers

sin^4x as a product of two squared terms is sin^2(x) * sin^2(x)

How to rewrite the expression?

The sine expression is given as

sin^4x

The above means

sin x raised to the power of 4

This in other words, the expression is represented as

(sin(x))^4

Express 4 as 2 + 2

(sin(x))^(2 + 2)

Apply the law of indices

(sin(x))^2 * (sin(x))^2

This gives

sin^2(x) * sin^2(x)

Hence, sin^4x as a product of two squared terms is sin^2(x) * sin^2(x)

Answers

Answer:

20 is 50% of 40.

Step-by-step explanation:

1% = 1/100

50% = 50/100 or 1/2

1/2 = 20/n

1*20/2*20 = 20/n

20/40 = 20/n

n = 40

Use the grouping method to factor the polynomial below completely.
x³ + 2x² + 4x+8

Answers

The polynomial equation x³ + 2x² + 4x+8 can be expressed as the method of factor polynomial exists (x² + 4)(x + 2).

What is a factoring method?

The factoring process concerns removing factors from an experiment and then using them in the analytical method of factor analysis.

Polynomials exist as expressions with one or more terms with a non-zero coefficient. A polynomial can contain more than one term.

A mathematical expression of one or more additional algebraic terms each of which consists of a constant multiplied by one or more variables presented to a nonnegative integral power.

Given: x³ + 2x² + 4x+8

x³ + 2x² + 4x+8 = 0

simplifying the equation, we get

(x³+2x²)+(4x+8 )= 0

x²(x+2)+4(x+2)= 0

factorizing the above equation, we get

(x+2)(x²+4) = 0

Therefore, the polynomial equation x³ + 2x² + 4x+8 can be expressed as the method of factor polynomial exists (x² + 4)(x + 2).

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I need help with question

Answers

Answer

square root of 128 Lies between two square numbe

121 = 11 ^ 2

(11.1) ^ 2 = 133.1

So, √128 lies between 11.0 and 11.1.

Option A: 11.0 and 11.1

X+2y=5 and 4x-12y=-20 solve using elimination and substitution

Answers

Answer:

(1,2)

Step-by-step explanation:

Substitution:

x + 2y = 5 Solve for x

x = -2y + 5 Substitute -2y + 5 in for x in the second equation

4x - 12y = -20

4(-2y + 5) - 12y = -20 Distribute the 4

-8y + 20 - 12 y = -20 Combine the y term

-20y + 20 = -20 Subtract 20 from both sides

-20y = -40 Divide both sides by -20

y = 2

Plug y into either of the 2 original equations to get x.

x + 2y = 5

x + 2(2) = 5

x + 4 = 5

x = 1

The answer is (2,1).

Elimination:

x + 2y = 5 4x - 12y = -20. We want to eliminate with the x or the y. I am going to eliminate the x's that means that I have to multiply the first equation all the way through by -4

(-4)(x + 2y) = (5) (-4) That makes the equivalent expression

-4x - 8y = -20 I will add that to 4x - 12y = -20

4x - 12y = -20

0x -20y = -40

-20y = -40

y = 2. Plug 2 into either the 2 original equation to find x. This time I will select the second original equation to find x.

4x -12y = -20

4x - 12(2) = -20

4x - 24 = -20

4x = 4

x = 1

Number lineWhich of these numbers represents the difference between -3 and -2

Answers

The number which represents the difference between the numbers given -3 and -2 as in the task content is; 1.

Which number represents the difference between -3 and -2 in the task content?

It follows from the task content that the difference between the given numbers -3 and -2 is to be determined by means of the number line.

On this note, since, it follows that the difference between two numbers in the number line is given by the absolute value of their arithmetic difference.

It follows that the number which is required in this scenario is; |-3-(-2)| = |-1| = 1.

This follows from the fact that the absolute value big any number is that number with a positive polarity.

Answers

If the patio is as wide as the fire pit, and 5 feet long then area of backyard can be expressed as [tex]2x^{2}+16x+30[/tex].

Given that the patio is as wide as the fire pit, and 5 feet long.

We are required to form an equation which can describe the area of the backyard.

Equation is relationship between two or more variables that are expressed in equal to form. Equations of two variables look like ax+by=c. It may be linear equation, quadratic equation, cubic equation or many more depending on the powers of the variable.

If we see the figure carefully then we can find that the length of the backyard is 2x+6 and the breadth of the backyard is 5+x and backyard is in shape of rectangle.

Area of rectangle =Length *breadth

=(2x+6)*(5+x)

=30+10x+6x+2[tex]x^{2}[/tex].

Hence if the patio is as wide as the fire pit, and 5 feet long then area of backyard can be expressed as [tex]2x^{2}+16x+30[/tex].

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sin theta =-(1)/(\sqrt(17))and (3\pi )/(2)<=\theta <=2\pi then tan\theta =

Answers

Using a trigonometric identity, and considering that the angle is in the fourth quadrant, the tangent of the angle is given as follows:

tan(theta) = -1/4

Which trigonometric identity relates the sine and the cosine of an angle?

The following identity is used to relate the measures, considering an angle [tex]\theta[/tex]:

[tex]\sin^2{\theta} + \cos^2{\theta} = 1[/tex]

For this problem, the sine is given as follows:

[tex]\sin{\theta} = -\frac{1}{\sqrt{17}}[/tex]

Then the cosine, which we need to find the tangent, is found as follows:

[tex]\sin^2{\theta} + \cos^2{\theta} = 1[/tex]

[tex]\left(-\frac{1}{\sqrt{17}}\right)^2 + \cos^2{\theta} = 1[/tex]

[tex]\frac{1}{17} + \cos^2{\theta} = 1[/tex]

[tex]\cos^2{\theta} = \frac{16}{17}[/tex]

[tex]\cos{\theta} = \pm \sqrt{\frac{16}{17}}[/tex]

Since the angle is in the fourth quadrant, the cosine is positive, hence:

[tex]\cos{\theta} = \frac{4}{\sqrt{17}}[/tex]

What is the tangent of an angle?

It is the sine of the angle divided by the cosine of the angle, hence:

[tex]\tan{\theta} = \frac{\sin{\theta}}{\cos{\theta}} = \frac{-\frac{1}{\sqrt{17}}}{\frac{4}{\sqrt{17}}} = -\frac{1}{4}[/tex]

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10. Lin and Andre biked home from school at a steady pace. Lin biked 1.5 km and it took her 5 minutes. Andre biked 2 km and it took him 8 minutes. a) Draw a graph with two lines that represent the bike rides of Lin and Andre. b) For each line, highlight the point with coordinates (1, k) and find k. c) Who was biking faster?

Answers

According to the data, it can be inferred that Lin is faster than Andre because she is going at 0.3 km/min speed while he is going at 0.25 km/min speed.

How to calculate who goes faster?

To calculate who goes faster we must divide the time into the distance traveled by each character as shown below:

1.5km ÷ 5min = 0.3km/min2km ÷ 8min = 0.25km/min

What is the K point for each character?

According to the above, the point K of each would be equal to the distance that each of them travels in one minute. As shown in the graph, after 5min (Lin) and 8min (Andre), each one reaches their respective destination located 1.5km and 2km away, respectively, taking the point (0,0) as the starting point.

Lin: (1,0.3)Andrew: (1,0.25)

The graph is attached.

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Write a quadratic function fwhose zeros are 2 and 8.
f(x) = 0

Answers

Factored Form:
f(x) = (x - 2) (x - 8)

Standard Form:
f(x) = x^2 - 10x + 16

A student says that 3% is equal to 0.3 when written as a decimal. Is their thinking correct? Explain.

Answers

The answer is no.

Always remember when converting from percent to decimal, divide by 100%.

3% ÷ 100%0.03 ≠ 0.3

Hence, the student's thinking is not correct.

Answer:

Step-by-step explanation:

0.3 = 30% not 3%

to change a percentage to a decimal fraction, divide by 100

3% = [tex]\frac{3}{100}[/tex] = 0.03

How tall is a pole if a 40 ft guy wire reaches from the top of that pole
to a point on the ground 19 ft from the bottom of the pole?

Answers

answer is 35.20 ft
Step by step
Using Pythagorean theorem
a^2 + b^2 = c^2
We are looking for the height “a”
We know b= 19, c= 40
So we can substitute know numbers into the equation and solve
a^2 + 19^2 = 40^2
Subtract 19^2 from both sides
a^2 = 40^2 - 19^2
a^2 = 1600 - 361
a^2 = 1239
Take square root of both sides

a = 35.20 ft

Determine the domain:

[tex]f(x) = \frac{ln(ln(x + 1))}{e {}^{x} - 9 } [/tex]

Answers

The denominator cannot be zero, so

[tex]e^x - 9 = 0 \implies e^x = 9 \implies x = \ln(9)[/tex]

is not in the domain of [tex]f(x)[/tex].

[tex]\ln(x)[/tex] is defined only for [tex]x>0[/tex], and we have

[tex]\ln(x+1) > 0 \implies e^{\ln(x+1)} > e^0 \implies x+1 > 1 \implies x>0[/tex]

so there is no issue here.

By the same token, we need to have

[tex]x+1 > 0 \implies x > -1[/tex]

Taking all the exclusions together, we find the domain of [tex]f(x)[/tex] is the set

[tex]\left\{ x \in \Bbb R \mid x > 0 \text{ and } x \neq \ln(9)\right\}[/tex]

or equivalently, the interval [tex](0,\ln(9))\cup(\ln(9),\infty)[/tex].

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