Find the perimeter of the triangle whose vertices are (−4,3), (−4,1), and (−5,−4). Write the exact answer. Do not round.

The three Pythagorean trigonometric identities, which I’m sure one can find in any Algebra-Trigonometry textbook, are as follows:

sin² θ + cos² θ = 1

tan² θ + 1 = sec² θ

1 + cot² θ = csc² θ

where angle θ is any angle in standard position in the xy-plane.

Consistent with the definition of an identity, the above identities are true for all values of the variable, in this case angle θ, for which the functions involved are defined.

The Pythagorean Identities are so named because they are ultimately derived from a utilization of the Pythagorean Theorem, i.e., c² = a² + b², where c is the length of the hypotenuse of a right triangle and a and b are the lengths of the other two sides.

This derivation can be easily seen when considering the special case of the unit circle (r = 1). For any angle θ in standard position in the xy-plane and whose terminal side intersects the unit circle at the point (x, y), that is a distance r = 1 from the origin, we can construct a right triangle with hypotenuse c = r, with height a = y and with base b = x so that:

c² = a² + b² becomes:

r² = y² + x² = 1²

y² + x² = 1

We also know from our study of the unit circle that x = r(cos θ) = (1)(cos θ) = cos θ and y = r(sin θ) = (1)(sin θ) = sin θ; therefore, substituting, we get:

(sin θ)² + (cos θ)² = 1

1.) sin² θ + cos² θ = 1 which is the first Pythagorean Identity.

Now, if we divide through equation 1.) by cos² θ, we get the second Pythagorean Identity as follows:

(sin² θ + cos² θ)/cos² θ = 1/cos² θ

(sin² θ/cos² θ) + (cos² θ/cos² θ) = 1/cos² θ

(sin θ/cos θ)² + 1 = (1/cos θ)²

(tan θ)² + 1 = (sec θ)²

2.) tan² θ + 1 = sec² θ

Now, if we divide through equation 1.) by sin² θ, we get the third Pythagorean Identity as follows:

(sin² θ + cos² θ)/sin² θ = 1/sin² θ

(sin² θ/sin² θ) + (cos² θ/sin² θ) = 1/sin² θ

1 + (cos θ/sin θ)² = (1/sin θ)²

1 + (cot θ)² = (csc θ)²

3.) 1 + cot² θ = csc² θ

An equation is a statement that two expressions are equal. The expressions on both sides of the equation can be made up of variables, constants, and mathematical operations.

For example, the equation:

[tex]2x + 5 = 11[/tex]

Has two expressions on either side of the equals sign. The expression on the left side is 2x + 5, which consists of the variable x, the constant 2, and the constant 5, combined using the mathematical operation of addition.

The expression on the right side is 11, which is a constant. The equation states that the two expressions are equal, which means that the value of x can be determined to be 3 by solving the equation.

Part A:

The value for x that is a solution to  [tex]2x - 5 = 3 is x = 4.[/tex]  

The value for x that is a solution to   [tex]2x - 5 > 3[/tex] is   [tex]x > 4[/tex] .

Part B:

The solution to  [tex]-2x - 5 = 3[/tex] is [tex]x = -4[/tex] .

The solution to   [tex]-2x - 5 > 3[/tex] is  [tex]x < -4[/tex]  .

A value for x that is a solution to  [tex]-2x - 5 = 3[/tex] is [tex]x = -4[/tex]  .

A value for x that is a solution to  [tex]-2x - 5 > 3[/tex] is [tex]x = -5[/tex]  .

Therefore, The value for x that is a solution to  [tex]2x - 5 > 3[/tex] is [tex]x > 4[/tex] . and A value for x that is a solution to [tex]-2x - 5 > 3[/tex]   is [tex]x = -5[/tex]  .

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Answer:

[tex]2679\frac{7}{15}[/tex]

Step-by-step explanation:

[tex]\frac{1}{3}\times 3.14\times 16^2 \times 10[/tex]

First thing we need to do is to evaluate 16^2.

[tex]16^2 = 16\times 16 =256[/tex]

Now, because multiplication is an associative and communitative property, the order that we multiply won't matter. I will rearrange terms to multiply in a more easier way, left to right.

[tex]\frac{1}{3}\times 10\times 256\times 3.14[/tex]

Lets multiply the fraction first. Multiply across, divide 10 by 3. 3R1 gives us:

[tex]3\frac{1}{3} \times 256 \times 3.14[/tex]

Now lets multiply 256. Same method as before.

[tex]853\frac{1}{3}\times 3.14[/tex]

Now, finally, the decimal. Lets convert it to a fraction.

[tex]3.14=\frac{314}{100}[/tex]

Now, replace the decimal in the expression.

[tex]853\frac{1}{3}\times \frac{314}{100}[/tex]

Same method as before. Through rigorous simplifying, we get:

[tex]2679\frac{7}{15}[/tex]

Answer:

The smallest possible average of five distinct positive even integers will occur if we choose the five smallest even integers. Since we want the integers to be distinct, we start with 2 and add the next four even integers:

2, 4, 6, 8, 10

The average of these five integers is:

(2 + 4 + 6 + 8 + 10) / 5 = 30 / 5 = 6

Therefore, the smallest possible average of five distinct positive even integers is 6, which is answer choice C.

Answer: The estimated product is about 10.

Step-by-step explanation:

To estimate the product, we can round 2 1/6 to 2 and 4 1/2 to 5. Then, we multiply 2 x 5 to get an estimated product of 10.

To find the exact product, we can use the following steps to multiply the two mixed numbers:

   Convert each mixed number to an improper fraction:

   2 1/6 = 13/6

   4 1/2 = 9/2

   Multiply the two fractions:

   (13/6) x (9/2) = (13 x 9) / (6 x 2) = 117/12

   Simplify the fraction, if possible:

   117/12 = 9 3/4

Therefore, the exact product of 2 1/6 x 4 1/2 is 9 3/4.

a) As per the given budget, the amount of interval that would be better for the city is 95% confidence interval.

b) The interval that be worse for the planners is depends on sample size

c) They achieve an interval estimate that would better serve their planning needs is depends on margin of error

d) The number of days worth of data should they collect to have confidence of estimating the true mean to 30 days

To obtain a better estimate, the city can create a confidence interval, which is a range of values that is likely to contain the true population mean with a certain degree of confidence.

However, there are also some disadvantages to using a confidence interval. The interval estimate may be wider than a point estimate, which means that the budget planners may have to allocate a larger budget to account for the uncertainty in the estimate.

To achieve a better interval estimate, the city could increase the sample size or reduce the variability of the data. Increasing the sample size reduces the margin of error and increases the precision of the estimate.

Finally, to determine how many days' worth of data the city should collect to estimate the true mean with a certain degree of confidence, the city would need to consider the desired level of precision, the variability of the data, and the desired level of confidence.

Typically, a larger sample size will provide a more accurate estimate, but this also depends on the variability of the data. In general, a sample size of at least 30 is recommended for a reasonably accurate estimate.

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The solution of equation of trigonometric functions, tan²(x) + tan(x) − 20 = 0, is equals to the x = 1.33 radians , 1.37 radians (x = arc tan(4) = 1.33 , x = arc tan(-5)= 1.37).

We know that trigonometric functions are periodic functions, solutions of trigonometric equations are then infinite and periodic. In these equations, it is essential to know the reduction formulas of each quadrant, which allows each angle in the first quadrant to be related to its corresponding angle in the other three quadrants. We have an equation which contains triagmometric function,

tan²(x) + tan(x) − 20 = 0 --(1) and we have to solve it. First we change the variable as y = tan(x). Rewrite the equation (1), y² + y - 20 = 0 --(2) which is an quadratic equation. The quadratic formula helps to solve the equation (2).

=> y = ( -1 ± √1 - 4(-20))/2

=> y = ( -1 ± √81)/2

=> y = ( -1 ± 9)/2

=> y = ( -1 + 9)/2 or (-1 -9)/2

=> y = 4, -5

Now x = tan⁻¹(y), and so, x = tan⁻¹(4) = 1.33 radians, or tan⁻¹(-5) = 1.37 radians. Hence, required value is ( 1.33, 1.37).

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Answer:Approximately

(0.58975,0.34781)

Step-by-step explanation:

If (x,y) is a point on the parabola, then the distance between (x,y) and (1,0) is:

√(x−1)2+(y−0)2=√x4+x2−2x+1

To minimize this, we want to minimize

f(x)=x4+x2−2x+1

The minimum will occur at a zero of:

f'(x)=4x3+2x−2=2(2x3+x−1)

graph{2x^3+x-1 [-10, 10, -5, 5]}

Using Cardano's method, find

x=3√14+√8736+3√14−√8736≅0.58975

y=x2≅0.34781

Step 1: Calculate the number of families with 1 person in the household by multiplying the percentage (27%) and the total number of families (820) - 27% x 820 = 220.4.

Step 2: Calculate the number of families with 2 people in the household by multiplying the percentage (33%) and the total number of families (820) - 33% x 820 = 271.6.

Step 3: Calculate the number of families with 3 people in the household by multiplying the percentage (24%) and the total number of families (820) - 24% x 820 = 196.8.

Step 4: Calculate the number of families with 4 people in the household by multiplying the percentage (10%) and the total number of families (820) - 10% x 820 = 82.

Step 5: Calculate the number of families with 5 people in the household by multiplying the percentage (4%) and the total number of families (820) - 4% x 820 = 32.8.

Step 6: Calculate the number of families with 6 or more people in the household by multiplying the percentage (2%) and the total number of families (820) - 2% x 820 = 16.4.

Step 7: Add the numbers calculated in Steps 1 - 6 to get the total number of families with at least 3 people in the household - 220.4 + 271.6 + 196.8

Event a has a 12/25 chance of happening provided that event b has already happened.

It is stated as a number between 0 and 1, where 0 denotes that the event is hypothetical and 1 denotes that it is unavoidable. Furthermore, probability can be stated as a percentage, with a range of 0% to 100%. By dividing the number of favourable outcomes by the entire number of possible possibilities, the probability of an event is determined. Many disciplines, such as statistics, economics, and physics, among others, utilise probability theory.

given

We may apply the conditional probability formula to determine p(a/b):

p(a and b) / p = p(a and b) (b)

Given that p(b) = 1/4 and p(a and b) = 3/25, we may enter these numbers in the formula as follows:

p(a/b) = (3/25) / (1/4)

p(a/b) = (3/25) * (4/1)

p(a/b) = 12/25

Event a has a 12/25 chance of happening provided that event b has already happened.

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To construct a confidence interval for sigma the population from which the sample was drawn must be normally distributed. -

It is not necessary to make the normalcy assumption in order to build a confidence interval for sigma. To employ the central limit theorem, however, the sample size must be adequate which ideally enables the use of the normal distribution to approximate the sampling distribution of the sample variance. Alternative techniques, like t-distribution, can be employed if total sample size is limited.

It's also important to remember that, regardless of sample size, the distribution of sample variance will be normal if population is known and recognizable to have a normally distributed population. As long as the sample size is appropriate, the sample variance may still be utilized to create a confidence range for sigma even if the population is not normally distributed.

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(a) The theoretical probability of getting a 5 or 6 is 1/5

(b) The experimental probability of getting a 5 or 6 is 1/5

(c) The true statement is the first statement.

A subfield of statistics known as probability studies random events and their likelihood of happening. It is calculated by dividing the number of favorable outcomes by the total number of possible outcomes and is used to make predictions and estimate the likelihood of future events.

(a) Assuming that the machine is fair, the theoretical probability of getting a 5 or 6 is:

P(5 or 6) = P(5) + P(6) = 1/10 + 1/10 = 1/5

(b) From the results, we can see that out of the 50 trials, there were 10 trials where the machine output a 5 or a 6.

The experimental probability of getting a 5 or 6 is:

P(5 or 6) = 10/50 = 1/5

(c) A large number of trials, might be a difference between the experimental and theoretical probabilities, but the difference should be small. This is because the theoretical probability is based on the assumption of a fair machine, while the experimental probability is based on actual results.

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Therefore , the solution of the given problem of unitary method comes out to be A, B, C, and D should be replaced by the frequencies 4, 5, 1, and 2 correspondingly.

It is possible to achieve the goal by utilizing already known variables, this widespread convenience, or all crucial elements from the first bishop malleable research that followed a particular methodology. Both vital components will surely miss the statement if the term assertion result does not occur; if it does, it will then be able to contact the entity once more.

Here,

Since two more people were asked, we need to add their responses to the tally and update the table:

Color Tally Frequency

Red IIII 4

Orange II 2

Yellow IIII 4

Green III 3

Blue IIII 4

Indigo I 1

Violet II 2

Unknown AAB 2

Since "Blue" and "Green," the two new responses, are not among the initial colors in the table, we can add them to the "Unknown" category. The revised total for "Unknown" is therefore AABBG.

Counting the tallies, we can update the frequency column:

Color Tally Frequency

Red IIII 4

Orange II 2

Yellow IIII 4

Green IIII 4

Blue IIIII 5

Indigo I 1

Violet II 2

Unknown AABBG 5

In the finished table, A, B, C, and D should be replaced by the frequencies 4, 5, 1, and 2 correspondingly.

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The set of all x-values that are at a distance of 13 or less from the number 8 in the interval notation is given by  [ -5, 21 ].

The distance between x and 8 is |x - 8|.

Find all the values of x such that |x - 8| ≤ 13.

This inequality can be rewritten as follow,

|x - 8| ≤ 13

⇒ -13 ≤ x - 8 ≤ 13

Now,

Adding 8 to all sides of the inequality we get,

⇒  -13 +  8 ≤ x - 8 + 8 ≤ 13 + 8

⇒ -5 ≤ x ≤ 21

Therefore, all the x-values which are at a distance of 13 or less from the number 8 represented in the interval notation as [ -5, 21 ].

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Answer:

4-. c = ±26

Step-by-step explanation:

c² = a² + b²

c² = 10² + 24²

c² = 100 + 576

c² = 676

√c² = √676

c = ± 26

Tom's fortnightly income is $3000.

In mathematics, an average is a measure that represents the central or typical value of a set of numbers. There are several types of averages commonly used, including the mean, median, and mode.

To calculate Tom's fortnightly income, we need to divide his yearly salary by the number of fortnights in a year:

Fortnightly income = Yearly salary / Number of fortnights in a year

Fortnightly income = $78000 / 26 = $3000

Therefore, Tom's fortnightly income is $3000.

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question - Calculate the  Tom's fortnightly income and yearly salary by the number of fortnights in a year .

The total amount payable at the end of the loan is Option 1: $369817.20 and Option 2: $233607.60.

APR, which stands for Annual Percentage Rate, refers to the yearly interest rate that is applied to loans. APR takes into account not just the interest rate but also any other expenditures or expenses related to the loan, including points or closing costs. APR offers a uniform approach to assess the costs and benefits of various lending solutions. Lenders are required by law to provide borrowers with the APR of a loan so they may make educated judgements about their borrowing alternatives.

The total payment for the loan is the monthly amount paid over the course of loan.

For option 1 the number of months are:

n = 30 (12) = 360

The total payable amount is:

$1027.27 (360) = $369817.20

For option 2 the number of months are:

n = 15 (12) = 180

The total payable amount is:

$1297.82 (180) = $233607.60

Hence, the total amount payable at the end of the loan is Option 1: $369817.20 and Option 2: $233607.60.

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The algebraic expression for the product of 9 and x is 9x. This can be expressed in steps as follows:

Step 1: Identify the values that are being multiplied together. In this case it is 9 and x.

Step 2: Write the two values side by side and place a multiplication sign between them.

Step 3: The algebraic expression for the product of 9 and x is then written as 9x.

The value of the limits are:

The highest degree terms in the numerator and denominator are both 3x^2.

So, as x approaches infinity, the expression behaves like √(3x^2)/√(3x^2) = 1.

Therefore, the limit evaluates to:

lim x → ∞√(9+3x²)/(2+7x) = lim x → ∞(√(3x²)/√(3x²))

lim x → ∞√(9+3x²)/(2+7x) = 1.

(b) The highest degree term in the numerator is 3x^2, while the highest degree term in the denominator is 7x.

Therefore, as x approaches negative infinity, the expression behaves like:

√(3x^2)/√(7x) = √(3/7)(x^2/x) = √(3/7)x.

Since the coefficient of x is positive, the expression approaches negative infinity as x approaches negative infinity.

Therefore, the limit evaluates to:

lim x→-∞ √(9+3x^2)/(2+7x) = lim x→-∞ √(3/7)x

lim x→-∞ √(9+3x^2)/(2+7x)  = -∞.

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Complete question:

Evaluate the following limits. If needed, enter 'INF' for ∞ and '-INF for -∞.

(a)

lim x → ∞√9+3x²/2+7x

(b)

lim x→-∞  √9+3x2/2+7x

Answer:

{(0,3),(1,2),(2,1),(3,0),4,-1)}

Step-by-step explanation:

{(0,3),(1,2),(2,1),(3,0),4,-1)}

x           y
0          3
1           2
2          1

3          0

4         -1

Answer:

1. x = 11

Step-by-step explanation:

1. x + 7 = 18

move 7 to right then change the sign
        x = 18 - 7
        x = 11
2. x - 13 = 15

move -13 to right then change the sign
        x = 15 + 13
        x = 28
3. 8x = 64
   8      8
divided by 8 both side
        x = 8
4. 5x-13-12=0 it this the correct given?
add same variable
   5x = 13 + 12
   5x = 25
   5      5

divided by 5 both side
         x = 5

Answer:

The farmer uses 221 pipes in total: 124 plastic pipes and 97 metal pipes.

Answer:

To find out how many pipes the farmer used in total, we simply add the number of plastic pipes to the number of metal pipes:

Total pipes = Plastic pipes + Metal pipes

Total pipes = 124 + 97

Total pipes = 221

Therefore, the farmer used a total of 221 pipes for the irrigation system.

A quadratic equation in the form [tex]x^2[/tex]+bx+c=0 that has the roots -8±4i is [tex]x^2[/tex] + 16x + 48 = 0

If the roots of a quadratic equation are -8+4i and -8-4i, then the factors of the quadratic equation are (x-(-8+4i))(x-(-8-4i))=0.

Simplifying this expression, we get:

(x+8-4i)(x+8+4i) = 0

Expanding this expression, we get:

[tex]x^2[/tex] + (8-4i+8+4i)x + (8-4i)(8+4i) = 0

Simplifying this expression, we get:

[tex]x^2[/tex] + 16x + ([tex]8^2[/tex] - [tex](4i)^2[/tex]) = 0

[tex]x^2[/tex] + 16x + 48 = 0

Therefore, the quadratic equation in the form [tex]x^2[/tex]+bx+c=0 that has the roots -8±4i is:

[tex]x^2[/tex] + 16x + 48 = 0

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From the figure 1. Two line segments are LA and EP. 2. Two rays are EC and AH. 3. Two lines are b and AP.

A ray is a segment of a line with a single endpoint and unlimited length in a single direction. A ray cannot be measured in terms of length.

The ends of a line segment are two. These endpoints are included, along with every point on the line that connects them. A segment's length can be measured, while a line's length cannot.

A line is a collection of points that extends in two opposing directions and is endlessly long and thin.

From the given figure we observe that,

1. Two line segments are LA and EP.

2. Two rays are EC and AH.

3. Two lines are b and AP.

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Answer: $1,775

Step-by-step explanation:

We can use a proportion to estimate the cost of having a 13 ft by 18 ft brick patio installed, based on the cost of a 15 ft by 20 ft patio:

(cost of 13 ft by 18 ft patio) / (cost of 15 ft by 20 ft patio) = (area of 13 ft by 18 ft patio) / (area of 15 ft by 20 ft patio)

The area of the 13 ft by 18 ft patio is 13 x 18 = 234 sq ft, and the area of the 15 ft by 20 ft patio is 15 x 20 = 300 sq ft.

So,

(cost of 13 ft by 18 ft patio) / ($2,275) = 234 / 300

Solving for the cost of the 13 ft by 18 ft patio, we get:

cost of 13 ft by 18 ft patio = ($2,275) x (234 / 300) = $1,775.25

Rounding to the nearest dollar, the estimated cost of having a 13 ft by 18 ft brick patio installed is $1,775.

Step-by-step explanation:

this is the answerrr

without calculator

The value of n that yields the smallest S_n element having an order of at least 100 is 101.

To find the minimum value of n for which the set S_n contains an element with an order equal to or greater than 100, the formula S_n = n!/r!(n-r)! can be used. This formula calculates the number of permutations in a set with n elements, where r elements are chosen at a time. By substituting r=100 into the formula, it is determined that n must be at least 101 to contain an element with an order of 100 or greater. Therefore, the smallest value of n for which S_n contains an element with an order of 100 or greater is 101.

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Complete question:

Find the smallest value of n that you can for which S_n has an element of order greater than or equal to 100

Modulus of rigidity (shear modulus) measures a material's rigidity in shear stress/strain. Polar moment of inertia measures the ability of a circular beam to resist torsion measured with J = 32 / (pi x D^4) , and torque is the twisting force on a specimen measured as  T = G x J x θ.

The modulus of rigidity or shear modulus, represented by G, is a measure of a material's rigidity when subjected to shear stress. Shear stress is the force applied perpendicular to the cross-sectional area of a material, while shear strain is the resulting deformation or twisting of the material.

The equation G = shear stress / shear strain is only valid in the elastic region of a material, where it can return to its original shape after the force is removed.

The polar moment of inertia, J, is a measure of a circular cross-section beam or specimen's resistance to torsion or twisting. A larger diameter of the beam results in a larger polar moment of inertia.

The equation J = 32 / (pi x D^4) is used to calculate the polar moment of inertia, where D is the diameter of the beam.

The torque in a circular cross-section beam or specimen is given by the equation T = G x J x θ, where G is the shear modulus, J is the polar moment of inertia, and theta is the angle of twist in radians.

The torque arm length and the applied force F are used to calculate the twisting force or torque in the specimen.

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The equation for the tangent line to the curve y = f(x) that passes through the point (1,1) is y = (x + 1)/2. Using this equation, we can estimate the value of f(1.2) to be approximately 1.1.

To find the equation for the tangent line to the curve y = f(x) that passes through the point (1,1), we first need to find the derivative of y with respect to x, which is given by:

dy/dx = xy/2

Next, we can use the point-slope form of the equation of a line to write the equation for the tangent line:

y - 1 = (x - 1)(dy/dx at (1,1))

dy/dx at (1,1) = (1*1)/2 = 1/2

y - 1 = (x - 1)(1/2)

2y - 2 = x - 1

2y - x = 1

y = (x + 1)/2

Therefore, the equation for the tangent line to the curve y = f(x) that passes through the point (1,1) is y = (x + 1)/2 .

To estimate the value of f(1.2), we can use the equation of the tangent line we just found:

y =  (x + 1)/2

At x = 1.2, the value of y can be estimated by substituting x = 1.2 into the equation of the tangent line:

y = (1.2 + 1)/2

y = 2.2/2

y = 1.1

Therefore, using the equation of the tangent line, we can estimate the value of f(1.2) to be approximately 1.1.

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--The question is incomplete, answering to the question below--

"let f be the function that satisfies the given differential equation dy/dx = xy/2. write an equation for the tangent line to the curve y = f(x) that pass through the point (1,1). Then use your tangent equation to estimate value of f(1.2)"

The function[tex]h(x) = -2x^4 + x^2 - 13; x = y+1,[/tex] can be simplified as [tex]h(y+1) = -2y^4 - 8y^3 - 10y^2 - 8y - 14.[/tex]

A function is a type of rule that produces one output for a single input. Source of the image: Alex Federspiel. This is illustrated by the equation y=x2. Any input for x results in a single output for y. Considering that x is the input value, we would state that y is a function of x.

To evaluate the function[tex]h(x) = -2x^4 + x^2 - 13[/tex] when x = y + 1, we can substitute y + 1 for x:

[tex]h(y+1) = -2(y+1)^4 + (y+1)^2 - 13[/tex]

Simplifying this expression involves some algebraic manipulation. We can start by expanding the fourth power using the binomial theorem:

[tex](y+1)^4 = y^4 + 4y^3 + 6y^2 + 4y + 1[/tex]

Substituting this expression into h(y+1), we get:

[tex]h(y+1) = -2(y^4 + 4y^3 + 6y^2 + 4y + 1) + (y^2 + 2y + 1) - 13[/tex]

Simplifying further, we get:

[tex]h(y+1) = -2y^4 - 8y^3 - 10y^2 - 8y - 14[/tex]

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