Find the minimum for this list of numbers
9
60
6
45
50
8
28
19
5
74
24
13
81
23
11

Mean - this is the average. Add up the numbers and divide by the count of the numbers in the data set.

11+13+14+16+17+18+20+20+21+24+25+29 = 228

There are 12 numbers, so divide by 12.

228/12 = 19. The mean is 19.

Median - - - this is the MIDDLE number. Your data is already in numerical order (hooray!) so there are 12 numbers. Since it's even, we'll take the average of the 2 middle numbers. The 6th number is 18 and the 7th number is 20. The average of these is 19. So the median is 19.

(Side note: yes, median and mean can be the same.)

Mode - - - this is repeat/most common numbers! 20 repeats. 20 is the mode.

Range - - - biggest number is 29, littlest is 11. The range is 29-11 = 18.

Answer:

12

Step-by-step explanation:

Given:

-8

+20°

Solution:

-8+20 = 12

Therefore the answer is 12

f(x) = 2x - 4

Then find

k(x) = f(x)

Now replace

x+ 3 = 2x -4

Solve for x

x + -2x = -4 - 3

. -x = -7

. x = 7

Now find y = f(x) = k(x)

If his average is 0.196, the probability of hitting is also 0.196.

So, in a total of 9 bats, the probability of hitting at least 3 can be calculated by first calculating the probability of hitting 0, 1 or 2 times.

To do so, we can use the following formula:

Where (9, x) is the binomial of 9 and x and p is the probability of hitting. So we have:

So the probability of hitting at least 3 times can be calculated as:

So the probability is 0.2516.

The table of values for the total cost is:

# of Rides, x    Total cost, y

0                             $8

1                              $11

2                               $14

3                               $17

4                                $20

5                                $23

The table of values create a linear function. This is because the table of value increases by a constant value.

The total cost is the sum of the amount the carnival charges, the number of rides and the cost per ride.

Total cost = cost to enter + (cost per ride x number of rides)

Total cost for 0 ride = $8 + ($3 x 0) = $8

Total cost for 1 ride = $8 + ($3 x 1) = $11

Total cost for 2 rides = $8 + ($3 x 2) = $14

Total cost for 3 rides = $8 + ($3 x 3) = $17

Total cost for 4 rides = $8 + ($3 x 4 ) = $20

Total cost for 5 rides = $8 + ($3 x 5) = $23

The table of values creates a linear function. This is because the total cost increases by a constant amount.

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Answer: b = 11.62

Step-by-step explanation:

We can use this formula to solve for b:

\(b^{2} =\) \(\sqrt{c^{2}-a^{2} }\)

\(b^{2} =\) \(\sqrt{12^{2}-3^2 }\)

\(b^2= \sqrt{144-9}\)

= 11.61895004

We can round that to 11.62.

Hope this helped!

Answer:

483

Step-by-step explanation:

A=1/2(a+b)h

A = 96.6

V = A x h

V = 96.6 x 5

V=483

The equation of the parallel line will be y = (2/3)x - 8/3.

Let the equation of the line be ax + by + c = 0. Then the equation of the parallel line that is parallel to the line ax + by + c = 0 is given as ax + by + d = 0.

The equation of the line is given below.

y - 4 = 2/3 (x - 3)

The equation of the line that is parallel to the given line will be written as,

y = (2/3)x + c

The equation of the line that passes through (1, -2), then we have

- 2= (2/3)(1) + c

- 2 = 2 /3 + c

c = - 8 / 3

Then the equation of the parallel line will be y = (2/3)x - 8/3.

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Check the picture below.

\(\textit{using the pythagorean theorem} \\\\ c^2=a^2+b^2 \qquad \begin{cases} c=\stackrel{hypotenuse}{4\sqrt{2}}\\ a=\stackrel{adjacent}{s}\\ b=\stackrel{opposite}{s}\\ \end{cases}\implies (4\sqrt{2})^2~~ = ~~s^2~~ + ~~s^2 \\\\\\ 4^2\sqrt{2^2}~~ = ~~2s^2\implies 16(2)=2s^2\implies 32=2s^2\implies \cfrac{32}{2}=s^2 \\\\\\ 16=s^2\implies \sqrt{16}=s\implies \boxed{4=s}\)

Answer:

Step-by-step explanation:

Answer:

5

Step-by-step explanation:

When you have a negative next to a negative, you can turn it into a positive. For example the -(-8) would now be +8. This is called the airplane strategy because it forms a airplane if you flip you sheet of paper or your device.

From here, we have -3+8.

Now we subtract since each value has a different sign and we get +5.

Hope this helps!! :)

Answer:

5

Step-by-step explanation:

-*-=+

-*+=-

+*+=+

-3-(-8)=5

-3+8=5

Answer:

  $142.32, profit on sale of the policy

Step-by-step explanation:

You want to know the expected value of a $280,000 life insurance policy sold for $270, if the probability the insured will live for the year is 0.999544.

The insurance company expects to have to pay the $280,000 death benefit for 0.000456 of the policies issued. That means their expected payout on any one policy is ...

  0.000456 × $280,000 = $127.68

The company gets a premium of $270 for the policy, so the expected value of the policy to the company is ...

  $270 -127.68 = $142.32

The expected value of the policy to the company is $142.32.

This represents its profit from sale of the policy.

__

Additional comment

Of course, the company has expenses related to the policy, perhaps including a commission to the agent selling it, and expenses related to handling claims. That is to say that not all of the difference between the premium and the average death benefit is actually profit. It is what might be called "contribution margin."

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Answer:

7.79

Step-by-step explanation:

A hundredth is the second number after the decimal. For example, in the number 100.234, the "3" would be the hundredth, the "2" would be the tenth, and the "4" would be the thousandth.

To round to the nearest hundredth, you have to look at the thousandth because when rounding, you look at the next number to decide whether to round up or down.

Since 8 is in the thousandth place in this question, and 8 is bigger than 5, you round up. So, you round the "8" in the hundredths place up to 9.

Therefore, you get 7.79.

Answer:

2

Step-by-step explanation:

( 3x + 6 ) ( 4x - 8 ) = 0

( 3x + 6 ) = 0 / ( 4x - 8 )

3x + 6 = 0

3x = 6

x = 6 / 3

x = 2

The value of the rational expression when x is equal to 4 is 2. The correct answer is option C: 2.

To find the value of the rational expression (x - 12)/(x - 8) when x is equal to 4, we substitute x = 4 into the expression:

[(4) - 12]/[(4) - 8]

Simplifying the numerator and denominator:

(4 - 12)/(-4)

Further simplifying the numerator:

(-8)/(-4)

Now, we can divide -8 by -4:

(-8)/(-4) = 2

So, when x is equal to 4, the value of the rational expression is 2.

Therefore, C is the right response.

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The correct option is B. The students knew whether or not music was playing while they were taking the exam

Given,

The class was split into two groups.

The first group took the midterm exam while background music was playing, and the second group took the exam without any background music.

In a blind experiment, any information that might affect the subjects is kept secret until the experiment is over. The pupils in this instance are aware of their group membership.

In order to answer this question, an experiment is carried out to see how music affects learning. One group of students studies while background music is playing, while the other group does not.

However, because of the background music playing here, the pupils are aware of which group they belong to. The experiment's outcomes are impacted by this.

Therefore the correct option is B. The students knew whether or not music was playing while they were taking the exam.

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Answer:

your answer is 21

Step-by-step explanation:

ffffffffffffff

\(\huge\text{Hey there!}\)

\(\mathsf{30\% \ of\ 70}\\\\\mathsf{=30\% \times 70}\\\\\mathsf{= \dfrac{30}{100}\times 70}\\\\\mathsf{= \dfrac{30}{100}\times \dfrac{70}{1}}\\\\\mathsf{= \dfrac{30\div10}{100\div 10}\times\dfrac{70}{1}}\\\\\mathsf{= \dfrac{3}{10}\times\dfrac{70}{1}}\\\\\mathsf{=\dfrac{3\times70}{10\times1}}\\\\\mathsf{= \dfrac{210}{10}}\\\\\mathsf{= \dfrac{210\div 10}{10\div10}}\\\\\mathsf{= \dfrac{21}{1}}\\\\\mathsf{= 21\div 1}\\\\\mathsf{= 21}\)

\(\huge\text{Therefore, your answer is: \boxed{\mathsf{Option\ D. \ 21}}}\huge\checkmark\)

\(\huge\text{Good luck on your assignment \& enjoy your day!}\)

~\(\frak{Amphitrite1040:)}\)

Answer:

The perimeter of this rectangle is 10 cm.

Answer:Start by dividing the numerator and denominator by 2.

12/32 = 6/16

Now that can also be divided by 2.

6/16 = 3/8

That's as far as we can go because 3/8 can not be divided any more.

We could have divided 12/32 by 4 and saved a step.

The required simplest form of the 12 / 32 is 3 / 8.

Given that,
To write 12 32 in simplest form.

In mathematics, it deals with numbers of operations according to the statements. There are four major arithmetic operators, addition, subtraction, multiplication and division,

The process in mathematics to operate and interpret the function to make the function or expression simple or more understandable is called simplifying and the process is called simplification.

Here,

To write 12 / 32 in simplest form,m

= 12 / 32
= 4 * 3   / 2 * 4 * 4
Eliminating common terms that are common in numerator and denominator.
= 3 / 8

Thus, the required simplest form of the 12 / 32 is 3 / 8.

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Answer:

Step-by-step explanation:

If you have to have online people help you just give up on that answer and guess

Answer:

(x,y)=(0,4)

Step-by-step explanation:

write as systems as equations, multiply both sides of the equation by -1, sum the equations vertically to eliminate at least one variable, divide both sides of the equation by -19, substitute the given value of x into the equation 5x+y=4, The possible solution of the system is the ordered pairs (x,y) =(0,4), then you're done. whew. ;)

Answer:

(a) Name: Multinomial distribution

Parameters: \(p_1 = 5\%\)   \(p_2 = 85\%\)   \(p_3 = 10\%\)  \(n = 20\)

(b) Range: \(\{(x,y,z)| x + y + z=20\}\)

(c) Name: Binomial distribution

Parameters: \(p_1 = 5\%\)      \(n = 20\)

\((d)\ E(x) = 1\)   \(Var(x) = 0.95\)

\((e)\ P(X = 1, Y = 17, Z = 3) = 0\)

\((f)\ P(X \le 1, Y = 17, Z = 3) =0.07195\)

\((g)\ P(X \le 1) = 0.7359\)

\((h)\ E(Y) = 17\)

Step-by-step explanation:

Given

\(p_1 = 5\%\)

\(p_2 = 85\%\)

\(p_3 = 10\%\)

\(n = 20\)

\(X \to\) High Slabs

\(Y \to\) Medium Slabs

\(Z \to\) Low Slabs

Solving (a): Names and values of joint pdf of X, Y and Z

Given that:

\(X \to\) Number of voids considered as high slabs

\(Y \to\) Number of voids considered as medium slabs

\(Z \to\) Number of voids considered as low slabs

Since the variables are more than 2 (2 means binomial), then the name is multinomial distribution

The parameters are:

\(p_1 = 5\%\)   \(p_2 = 85\%\)   \(p_3 = 10\%\)  \(n = 20\)

And the mass function is:

\(f_{XYZ} = P(X = x; Y = y; Z = z) = \frac{n!}{x!y!z!} * p_1^xp_2^yp_3^z\)

Solving (b): The range of the joint pdf of X, Y and Z

Given that:

\(n = 20\)

The number of voids (x, y and z) cannot be negative and they must be integers; So:

\(x + y + z = n\)

\(x + y + z = 20\)

Hence, the range is:

\(\{(x,y,z)| x + y + z=20\}\)

Solving (c): Names and values of marginal pdf of X

We have the following parameters attributed to X:

\(p_1 = 5\%\) and \(n = 20\)

Hence, the name is: Binomial distribution

Solving (d): E(x) and Var(x)

In (c), we have:

\(p_1 = 5\%\) and \(n = 20\)

\(E(x) = p_1* n\)

\(E(x) = 5\% * 20\)

\(E(x) = 1\)

\(Var(x) = E(x) * (1 - p_1)\)

\(Var(x) = 1 * (1 - 5\%)\)

\(Var(x) = 1 * 0.95\)

\(Var(x) = 0.95\)

\((e)\ P(X = 1, Y = 17, Z = 3)\)

In (b), we have: \(x + y + z = 20\)

However, the given values of x in this question implies that:

\(x + y + z = 1 + 17 + 3\)

\(x + y + z = 21\)

Hence:

\(P(X = 1, Y = 17, Z = 3) = 0\)

\((f)\ P{X \le 1, Y = 17, Z = 3)\)

This question implies that:

\(P(X \le 1, Y = 17, Z = 3) =P(X = 0, Y = 17, Z = 3) + P(X = 1, Y = 17, Z = 3)\)

Because

\(0, 1 \le 1\) --- for x

In (e), we have:

\(P(X = 1, Y = 17, Z = 3) = 0\)

So:

\(P(X \le 1, Y = 17, Z = 3) =P(X = 0, Y = 17, Z = 3) +0\)

\(P(X \le 1, Y = 17, Z = 3) =P(X = 0, Y = 17, Z = 3)\)

In (a), we have:

\(f_{XYZ} = P(X = x; Y = y; Z = z) = \frac{n!}{x!y!z!} * p_1^xp_2^yp_3^z\)

So:

\(P(X=0; Y=17; Z = 3) = \frac{20!}{0! * 17! * 3!} * (5\%)^0 * (85\%)^{17} * (10\%)^{3}\)

\(P(X=0; Y=17; Z = 3) = \frac{20!}{1 * 17! * 3!} * 1 * (85\%)^{17} * (10\%)^{3}\)

\(P(X=0; Y=17; Z = 3) = \frac{20!}{17! * 3!} * (85\%)^{17} * (10\%)^{3}\)

Expand

\(P(X=0; Y=17; Z = 3) = \frac{20*19*18*17!}{17! * 3*2*1} * (85\%)^{17} * (10\%)^{3}\)

\(P(X=0; Y=17; Z = 3) = \frac{20*19*18}{6} * (85\%)^{17} * (10\%)^{3}\)

\(P(X=0; Y=17; Z = 3) = 20*19*3 * (85\%)^{17} * (10\%)^{3}\)

Using a calculator, we have:

\(P(X=0; Y=17; Z = 3) = 0.07195\)

So:

\(P(X \le 1, Y = 17, Z = 3) =P(X = 0, Y = 17, Z = 3)\)

\(P(X \le 1, Y = 17, Z = 3) =0.07195\)

\((g)\ P(X \le 1)\)

This implies that:

\(P(X \le 1) = P(X = 0) + P(X = 1)\)

In (c), we established that X is a binomial distribution with the following parameters:

\(p_1 = 5\%\)      \(n = 20\)

Such that:

\(P(X=x) = ^nC_x * p_1^x * (1 - p_1)^{n - x}\)

So:

\(P(X=0) = ^{20}C_0 * (5\%)^0 * (1 - 5\%)^{20 - 0}\)

\(P(X=0) = ^{20}C_0 * 1 * (1 - 5\%)^{20}\)

\(P(X=0) = 1 * 1 * (95\%)^{20}\)

\(P(X=0) = 0.3585\)

\(P(X=1) = ^{20}C_1 * (5\%)^1 * (1 - 5\%)^{20 - 1}\)

\(P(X=1) = 20 * (5\%)* (1 - 5\%)^{19}\)

\(P(X=1) = 0.3774\)

So:

\(P(X \le 1) = P(X = 0) + P(X = 1)\)

\(P(X \le 1) = 0.3585 + 0.3774\)

\(P(X \le 1) = 0.7359\)

\((h)\ E(Y)\)

Y has the following parameters

\(p_2 = 85\%\)  and    \(n = 20\)

\(E(Y) = p_2 * n\)

\(E(Y) = 85\% * 20\)

\(E(Y) = 17\)

1. The price of a pair of gloves (x) is $8.50, and the price of a hat (y) is $6.50.

2. Aaliyah invested $7,200 in the mutual fund that rose 6% and $2,800 in the mutual fund that rose 9%.

1. Let's assume the price of a pair of gloves is represented by 'x' and the price of a hat is represented by 'y'.

From the given information, we can establish the following equations based on Cody and Tor's purchases:

Equation 1: 2x + 4y = 43.00 (Cody's purchase)

Equation 2: 2x + 2y = 30.00 (Tor's purchase)

To solve this system of equations, we can use a method like substitution or elimination.

Using the substitution method, we can rearrange Equation 2 to express x in terms of y:

2x = 30.00 - 2y

x = 15.00 - y

Now, we substitute this value of x into Equation 1:

2(15.00 - y) + 4y = 43.00

30.00 - 2y + 4y = 43.00

2y = 43.00 - 30.00

2y = 13.00

y = 6.50

Substituting this value of y back into Equation 2 to find x:

2x + 2(6.50) = 30.00

2x + 13.00 = 30.00

2x = 30.00 - 13.00

2x = 17.00

x = 8.50

Therefore, the price of a pair of gloves (x) is $8.50, and the price of a hat (y) is $6.50.

2. Let's assume Aaliyah invested an amount represented by 'x' in the mutual fund that rose 6% and an amount represented by 'y' in the mutual fund that rose 9%.

From the given information, we can establish the following equations based on the rise in investment:

Equation 1: 0.06x + 0.09y = 684 (rise in investment)

Equation 2: x + y = 10,000 (total investment)

To solve this system of equations, we can use a method like substitution or elimination.

Using the substitution method, we can rearrange Equation 2 to express x in terms of y:

x = 10,000 - y

Now, we substitute this value of x into Equation 1:

0.06(10,000 - y) + 0.09y = 684

600 - 0.06y + 0.09y = 684

0.03y = 84

y = 2800

Substituting this value of y back into Equation 2 to find x:

x + 2800 = 10,000

x = 10,000 - 2800

x = 7200

Therefore, Aaliyah invested $7,200 in the mutual fund that rose 6% and $2,800 in the mutual fund that rose 9%.

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Answer:

x=26

Step-by-step explanation:

(x+6)2=64

x+6=32

x=26

Answer:

Here is the solution. Mark as brainlist please.

Answer:

3/18 or 1/6 in simplest form.

Step-by-step explanation:

We do (Y2-Y1)/(X2-X1).

(0--3)/(14--4)

When you subtract a negative, the sign turns positive.

(0+3)/14+4) =3/18

if you need to simply your answer it would be 1/6 because both the numerator and denominator are both divisible by 3.

Answer:

The correct answer is B, I took the test :)

You just turn the percentages into decimals

The probability distribution table is formed clearly in the answer part.

Probability is the branch of Mathematics that deals with the measurement of the chance of occurrence of a random event.

The probability of any event always lie in the close interval of 0 and 1 [0,1].

The probability for all the elements in the sample space can be shown in one table known as distribution table.

Suppose X be the event of choosing a major.

Then, its probability can be given as P(X).

Here, the percent value of a given event is equivalent to their probability.

It can be represented in the form of table as,

X (Major chosen by student)                              P(X) (Probability)

    Math                                                                    3% = 0.03

   Nursing                                                                22% = 0.22

  Psychology                                                           16% = 0.16

 Criminal justice                                                      11% = 0.11

   Business                                                              48% = 0.48

This table is known as the probability distribution table.

Hence, the probability distribution table is formed by calculating the probability for each of the events.

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The amount of money that she will have in total at the end of 4 years would be = $61,120

The total amount of money in Lauren's savings (P)= $40,000

The interest rate (R)= 13.2%

The number of years involved(T) = 4 years

The simple interest = principal×time× rate/100

simple interest = 40,000×4×13.2 /100

= 2112000/100

= $21,120

Therefore the total amount of money after 4 years = $40,000+$21,120= $61,120

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If the triangle on the left had the same area as the circle on the right, it would require more resources and potentially be more unstable than the current configuration.

If the triangle on the left had the same area as the circle on the right, it would mean that the triangle would have to be larger than its current size. This is because the area of a circle is determined by the formula A=πr^2, where r is the radius of the circle.

Therefore, if the area of the circle is equal to the area of the triangle, the radius of the circle would have to be equal to the height of the triangle, and the base of the triangle would need to be wider.

This would result in a larger triangle with a greater surface area than the current triangle. The larger triangle would also have a longer perimeter, which would make it more difficult to enclose and would require more material to construct.

Additionally, the larger triangle would have a higher center of gravity, which could make it more difficult to balance and more prone to tipping over.

Overall, if the triangle on the left had the same area as the circle on the right, it would require more resources and potentially be more unstable than the current configuration.

It is important to consider both the area and the shape of an object when determining its practicality and effectiveness in a given situation.

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Answer:

(i) x ≤ 1

(ii) ℝ except 0, -1

(iii) x > -1

(iv) ℝ except π/2 + nπ, n ∈ ℤ

Step-by-step explanation:

(i) The number inside a square root must be positive or zero to give the expression a real value. Therefore, to solve for the domain of the function, we can set the value inside the square root greater or equal to 0, then solve for x:

\(1-x \ge 0\)

\(1 \ge x\)

\(\boxed{x \le 1}\)

(ii) The denominator of a fraction cannot be zero, or else the fraction is undefined. Therefore, we can solve for the values of x that are NOT in the domain of the function by setting the expression in the denominator to 0, then solving for x.

\(0 = x^2+x\)

\(0 = x(x + 1)\)

\(x = 0\)     OR     \(x = -1\)

So, the domain of the function is:

\(R \text{ except } 0, -1\)

(ℝ stands for "all real numbers")

(iii) We know that the value inside a logarithmic function must be positive, or else the expression is undefined. So, we can set the value inside the log greater than 0 and solve for x:

\(x+ 1 > 0\)

\(\boxed{x > -1}\)

(iv) The domain of the trigonometric function tangent is all real numbers, except multiples of π/2, when the denominator of the value it outputs is zero.

\(\boxed{R \text{ except } \frac{\pi}2 + n\pi} \ \text{where} \ \text{n} \in Z\)

(ℤ stands for "all integers")

Answer:

(i)  x ≤ 1

(ii)  All real numbers except x = 0 and x = -1.

(iii)  x > -1

(iv)  All real numbers except x = π/2 + πn, where n is an integer.

Step-by-step explanation:

The domain of a function is the set of all possible input values (x-values).

\(\hrulefill\)

\(\textsf{(i)} \quad f(x)=\sqrt{1-x}\)

For a square root function, the expression inside the square root must be non-negative. Therefore, for function f(x), 1 - x ≥ 0.

Solve the inequality:

\(\begin{aligned}1 - x &\geq 0\\\\1 - x -1 &\geq 0-1\\\\-x &\geq -1\\\\\dfrac{-x}{-1} &\geq \dfrac{-1}{-1}\\\\x &\leq 1\end{aligned}\)

(Note that when we divide or multiply both sides of an inequality by a negative number, we must reverse the inequality sign).

Hence, the domain of f(x) is all real numbers less than or equal to -1.

\(\boxed{\begin{aligned} \textsf{Inequality notation:} \quad &x \leq 1\\\textsf{Interval notation:} \quad &(-\infty, 1]\\\textsf{Set-builder notation:} \quad &\left\{x \in \mathbb{R}\left|\: x \leq 1 \right\} \end{aligned}}\)

\(\hrulefill\)

\(\textsf{(ii)} \quad g(x) = \dfrac{1}{x^2 + x}\)

To find the domain of g(x), we need to identify any values of x that would make the denominator equal to zero, since division by zero is undefined.

Set the denominator to zero and solve for x:

\(\begin{aligned}x^2 + x &= 0\\x(x + 1) &= 0\\\\\implies x &= 0\\\implies x &= -1\end{aligned}\)

Therefore, the domain of g(x) is all real numbers except x = 0 and x = -1.

\(\boxed{\begin{aligned} \textsf{Inequality notation:} \quad &x < -1 \;\;\textsf{or}\;\; -1 < x < 0 \;\;\textsf{or}\;\; x > 0\\\textsf{Interval notation:} \quad &(-\infty, -1) \cup (-1, 0) \cup (0, \infty)\\\textsf{Set-builder notation:} \quad &\left\{x \in \mathbb{R}\left|\: x \neq 0,x \neq -1 \right\} \end{aligned}}\)

\(\hrulefill\)

\(\textsf{(iii)}\quad h(x) = \log_7(x + 1)\)

For a logarithmic function, the argument (the expression inside the logarithm), must be greater than zero.

Therefore, for function h(x), x + 1 > 0.

Solve the inequality:

\(\begin{aligned}x + 1 & > 0\\x+1-1& > 0-1\\x & > -1\end{aligned}\)

Therefore, the domain of h(x) is all real numbers greater than -1.

\(\boxed{\begin{aligned} \textsf{Inequality notation:} \quad &x > -1\\\textsf{Interval notation:} \quad &(-1, \infty)\\\textsf{Set-builder notation:} \quad &\left\{x \in \mathbb{R}\left|\: x > -1\right\} \end{aligned}}\)

\(\hrulefill\)

\(\textsf{(iv)} \quad k(x) = \tan x\)

The tangent function can also be expressed as the ratio of the sine and cosine functions:

\(\tan x = \dfrac{\sin x}{\cos x}\)

Therefore, the tangent function is defined for all real numbers except the values where the cosine of the function is zero, since division by zero is undefined.

From inspection of the unit circle, cos(x) = 0 when x = π/2 and x = 3π/2.

The tangent function is periodic with a period of π. This means that the graph of the tangent function repeats itself at intervals of π units along the x-axis.

Therefore, if we combine the period and the undefined points, the domain of k(x) is all real numbers except x = π/2 + πn, where n is an integer.

\(\boxed{\begin{aligned} \textsf{Inequality notation:} \quad &\pi n\le \:x < \dfrac{\pi }{2}+\pi n\quad \textsf{or}\quad \dfrac{\pi }{2}+\pi n < x < \pi +\pi n\\\textsf{Interval notation:} \quad &\left[\pi n ,\dfrac{\pi }{2}+\pi n\right) \cup \left(\dfrac{\pi }{2}+\pi n,\pi +\pi n\right)\\\textsf{Set-builder notation:} \quad &\left\{x \in \mathbb{R}\left|\: x \neq \dfrac{\pi}{2}+\pi n\;\; (n \in\mathbb{Z}) \right\}\\\textsf{(where $n$ is an integer)}\end{aligned}}\)