11. Reason Inductively Complete the statement: If 8 = 2, then
m-12=. Explain.

Answer:

Each of these functions is a transformation of the parent function f(x) = |x|. The transformations include shifting the graph up or down, stretching or compressing the graph vertically, reflecting the graph across the x-axis, and shifting the graph left or right. The vertex of each graph is located at a different point.

Step-by-step explanation:

- The function f(x) = |x| - 3 is a transformation of the parent function f(x) = |x|. The "-3" at the end of the function shifts the graph 3 units down. This means that the vertex, or lowest point, of the graph is at (0, -3) instead of (0, 0).

- The function f(x) = -|x - 4| is also a transformation of the parent function f(x) = |x|. The negative sign in front of the absolute value function reflects the graph across the x-axis. The "-4" inside the absolute value function shifts the graph 4 units to the right. This means that the vertex of the graph is at (4, 0) instead of (0, 0).

- The function f(x) = (1/3)|x| is a transformation of the parent function f(x) = |x|. The "1/3" in front of the absolute value function stretches the graph vertically by a factor of 1/3. This means that the graph is narrower and closer to the x-axis than the parent function. However, because the absolute value function is symmetrical, the graph is still centered at (0, 0).

- The function f(x) = -2|x - 1| is also a transformation of the parent function f(x) = |x|. The negative sign in front of the absolute value function reflects the graph across the x-axis. The "-1" inside the absolute value function shifts the graph 1 unit to the right. The "2" in front of the absolute value function stretches the graph vertically by a factor of 2. This means that the graph is narrower and closer to the x-axis than the parent function, and it is also reflected across the x-axis. The vertex of the graph is at (1, 0).

The evaluations are:

i. 11!  = 39916800

ii. 24! = 2.045 x 10^37

Expansion by factorial is a method of expanding a given number from 1 to the number given. An example is given thus;

5! = 5 x 4 x 3 x 2 x 1!

   = 120

To evaluate the given expressions;

a. 11! = 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1!

       = 39916800

Also,

b. 24! = 24 x 23 x 22 x 20 x ........ x 5 x 4 x 3 x 2 x 1!

          = 204484017332398 x 10 ^23

          = 2.045 x 10^37

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Answer: 34.9

Step-by-step explanation: 100 - 65.1

From Table 2, we can see that Georgia's income tax rate is 7%.

To find the amount of state income taxes Anthony will owe, we can multiply his income by the tax rate:

State income tax = (Anthony's income) x (Georgia's tax rate)

State income tax = $65,750 x 0.07

State income tax = $4,602.50

Therefore, Anthony will owe $4,602.50 in state income taxes. To find out how much money he will have left after he pays these taxes, we can subtract the tax amount from his income:

Money left after paying state income taxes = Anthony's income - State income tax

Money left after paying state income taxes = $65,750 - $4,602.50

Money left after paying state income taxes = $61,147.50

Therefore, Anthony will have $61,147.50 left after he pays state income taxes.

The [tex]f^{-1}[/tex] inverse of f  of each

1) (x + 1)/2

2) [tex]\sqrt{\frac{x+1}{4} } \\[/tex]

3) 3 - 4x

4) [tex](x+5)^{1/3}[/tex]

5) [tex]\sqrt{x + 4}[/tex]

1) function f(x) = 3x + 1

let f(x) = y

y = 3x + 1

y - 1 = 3x

(y - 1)/3 = x

x obtained is [tex]f^{-1}[/tex]

and y will be x

[tex]f^{-1}[/tex] = (x-1)/3

2) equation f(x) = 4x² - 1

let f(x) = y

y = 4x² - 1

y + 1  = 4x²

(y + 1 )/4 = x²

x = [tex]\sqrt{\frac{x + 1}{4} }[/tex]

x obtained is [tex]f^{-1}[/tex]

and y will be x

[tex]f^{-1}[/tex] =  [tex]\sqrt{\frac{x + 1}{4} }[/tex]

similarly,

3) f(x) = 3-x/4

y = 3-x/4

4y = 3 - x

x = 3 - 4x

[tex]f^{-1}[/tex] =  3- 4x

4) y = x³ - 5

y + 5 = x³

x = [tex](y + 5)^{1/3}[/tex]

[tex]f^{-1}[/tex] = [tex](x+5)^{1/3}[/tex]

5) y = x² - 4

y + 4 = x²

x = √y + 4

[tex]f^{-1} = \sqrt{x + 4}[/tex]

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The value of x it the nearest tenth of degree is 25.6°

The trigonometric functions are real functions which relate an angle of a right-angled triangle to ratios of two side lengths.

Sin(tetha) = opp/ hyp

cos( tetha) = adj/ hyp

tan( tetha) = opp/ adj

Here, the hypotenuse is 92 and adjascent to the to the angle x is 63

therefore to calculate x we use cosine function

cos x = 83/92

cos x = 0.902

x = cos^-1( 0.902)

x = 25.6° ( nearest tenth)

therefore the value of x is 25.6°

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The result of the synthetic division is:4x⁵+17x⁴+14x³-3x²+2 = (4x⁴ + 5x³ - x²) (x + 3) + 56

what is synthetic division ?

Synthetic division is a simplified method of dividing a polynomial by a linear factor. It is commonly used in algebra to find the quotient and remainder when dividing a polynomial by a linear factor of the form (x-a).

In the given question,

To use synthetic division to rewrite the fraction, we first set up the problem like this:

   -3 | 4    17    14    -3    0    2

       |     -12    -15    3   0

       +----------------------------------

          4     5    -1     0    0    2

The coefficients of the polynomial are written in descending order of powers of x, with any missing terms replaced by zeros. The divisor, x + 3, is written on the left, and the first coefficient of the polynomial is written on the top of the vertical line.

We then proceed to perform synthetic division as follows:

Bring down the first coefficient, which is 4, and write it under the horizontal line.

Multiply the divisor, -3, by the number we just brought down, which gives -12. Write this number under the next coefficient, 17.

Add the two numbers, 17 and -12, to get 5. Write this number under the next coefficient, 14.

Multiply the divisor, -3, by 5, which gives -15. Write this number under the next coefficient, -3.

Add the two numbers, -3 and -15, to get -18. Write this number under the next coefficient, 0.

Multiply the divisor, -3, by -18, which gives 54. Write this number under the next coefficient, 2.

Add the two numbers, 2 and 54, to get 56. This is the remainder, which we write above the divisor.

The result of the synthetic division is:

4x⁵+17x⁴+14x³-3x²+2 = (4x^4 + 5x³ - x²) (x + 3) + 56

Therefore, we have rewritten the fraction in the desired form:

4x⁵+17x⁴+14x³-3x²+2

---------------------- = 4x⁴ + 5x³ - x² + 56/(x+3)

         x+3

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If line A is parallel to line B then the value of x is 18.

Given that a line A is parallel to line B

We have to find the value of x

3x+7x=180

Combine the like terms

10x=180

Divide both sides by 10

value x=18

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Answer:

Step-by-step explanation:

Typically you see f(x)=something like you do up in the original equation.

that is your function/graph.

a) f(0)  means plug in 0 for x

   f(0) =  (-0)³-0²-0+13

   f(0) = 13

b)  f(2) =(-2)³-2²-2+13          >simplify the exponenents first

                                               (-2)³ = (-2)(-2)(-2)= -8

    f(2) = -8-4-2+13               >simplify work from left to right

    f(2) = -12-2+13

    f(2) = -14+13

    f(2) = -1

c)  f(-2) = [-(-2)]³-(-2)²-(-2)+13              >multiply - and -  for  (-(-2))³

    f(-2) = (2)³-(-2)²-(-2)+13                  >simplify exponents

    f(-2) = 8-4-(-2)+13          

    f(-2) = 4-(-2)+13

    f(-2) = 6+13

    f(-2) = 18

d)  means add f(1) and f(-1)

f(1) +f(-1) = (-1)³-1²-1+13 +[-(-1)]³-(-1)²-(-1)+13  

f(1) +f(-1) = -1-1-1+13 +[1]³-1-(-1)+13

f(1) +f(-1) = -1-1-1+13 +1-1+1+13            

f(1) +f(-1) = -2-1+13 +1-1+1+13

f(1) +f(-1) = -3 +13 +1-1+1+13

f(1) +f(-1) = 10+1-1+1+13

f(1) +f(-1) = 11 -1+1+13

f(1) +f(-1) = 10 +1 +13

f(1) +f(-1) = 24

The car will be worth half its original value after 4 years

A graph for straight-line depreciation for a car is given.

a

The number of years the car totally depreciates is 8 years as seen in the graph which is x -intercept.

Step 3/7

The model makes straight-line depreciation. In a straight line depreciation equation, the intercepts are

(0, maximum car value) and (maximum lifespan,0)

Let y represent the value of the car at any time during its lifetime. The minimum y -value is zero dollars and the maximum y -value is the purchase price of $25,600

The intercepts are (0, 25, 600) and (8,0)

Determine the slope of the equation.

Two points determine a line, so only two points are needed to determine the slope of a line.

Let the coordinates of the -intercept be the first point. That is, (xi,yi) = (0, 25, 600)

Let the coordinates of the -intercept be the second point. That is (x2, y2)= (8,0)

Use the slope ratio as shown below;

The slope of the depreciation equation is.

-3200

The general form for the equation of a straight line is:

y = mx + b

Where m represents the slope of the line and b represents the y-intercept.

The slope is -3,200 and the -intercept is 25,600 as obtained.

Therefore, the straight line depreciation equation is:

y = -3,200x + 25,600

Therefore, the value of the car after 4 years is $12,800

b.

Use the depreciation equation and substitute x=5 years to find the car worth after 5 years.

Therefore, the value of the car after 5 years is $9,600

c.

The half value of the car is half of $25,600 which is $12,800

The car will be worth half its original value after 4 years

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A. The expression that represents the total amount of canned food they have collected so far is: 4x² + 2xy + 3

B. Number of canned food still remaining to be collected is: 2x² - 4xy + 5

A polynomial expression is a mathematical expression consisting of variables and coefficients that involves only the operations of addition, subtraction, and multiplication of non-negative integer exponents.

Part A: To find the expression that represents the total amount of canned food they have collected so far, add the expression for each person given together.

3x² - 2 + x² + 2xy + 5

Combine like terms:

4x² + 2xy + 3

Part B: Number of canned food still remaining to be collected is calculated below.

(6x² - 2xy + 8) - (4x² + 2xy + 3)

Open bracket:

6x² - 2xy + 8 - 4x² - 2xy - 3

Combine like terms:

2x² - 4xy + 5

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There were 64 players at the start of the tournament.

To determine the number of players at the start of the tournament

We need to work backwards from the final round to the first round.

In the final round, there will be only two players remaining since the winner of that match will be declared the overall champion.

In the penultimate (fifth) round, there will be four players remaining.

These four players must have come from the previous round, where there were eight players.

Continuing this pattern, in the fourth round, there will be eight players, and in the third round, there will be 16 players.

In the second round, there will be 32 players, and in the first (initial) round, there will be 64 players.

Therefore, there were 64 players at the start of the tournament.

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a) The probability that a randomly selected student is taking a Math class or an English class is 0.82.

b) The probability that a randomly selected student is taking neither a math class nor an English class is 0.18.

Probability refers to the chance or likelihood that an expected success, event, or outcome occurs from many possible successes, events, or outcomes.

Probability is represented as a fractional value using decimals, fractions, or percentages.

The percentage of students taking a math class =79%

The percentage of students taking an English class = 70%

The percentage of students taking both classes = 67%

Let the event that a student is taking a math class = m

Let the event that a student is taking an English class = e

The probability of m is p(m) = 0.79

The probability of e is p(e) = 0.70

The probability of m and e is p(m and e) = 0.67

The probability that a randomly selected student is taking a math class or an English class = 0.82 (0.79 + 0.70 - 0.67)

The probability that a randomly selected student is taking neither a math class nor an English class = 0.18 (1 - 0.82)

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Answer:

5)

The inverse of the function f(x) = x^7 can be found by following these steps:

Step 1: Replace f(x) with y. The equation becomes y = x^7.

Step 2: Interchange x and y in the equation, so it becomes x = y^7.

Step 3: Solve the equation for y by taking the seventh root of both sides. This yields y = x^(1/7).

Therefore, the inverse function of f(x) = x^7 is g(x) = x^(1/7), which maps any given value of x to its seventh root.

It's important to note that the domain and range of the inverse function are the opposite of those of the original function. The domain of the inverse function is all real numbers, while the range is only positive real numbers. The domain of the original function is all real numbers, while the range is also all real numbers.

6)

To find the inverse of the function f(x) = (-2/5)x^3, we can follow these steps:

Step 1: Replace f(x) with y. The equation becomes y = (-2/5)x^3.

Step 2: Solve the equation for x in terms of y.

Multiply both sides by -5/2:

(-5/2) y = x^3

Take the cube root of both sides:

x = [(-5/2) y]^(1/3)

Step 3: Replace x with f^-1(y) to obtain the inverse function.

f^-1(y) = [(-5/2) y]^(1/3)

Therefore, the inverse function of f(x) = (-2/5)x^3 is f^-1(y) = [(-5/2) y]^(1/3).

It is important to note that the domain and range of the inverse function are the opposite of those of the original function. The domain of the inverse function is all real numbers, while the range is also all real numbers. The domain of the original function is all real numbers, while the range is only negative real numbers if x is negative and only positive real numbers if x is positive.

Therefore, the correct answer is option C that is LM is reflected over the y-axis to L'M.

In mathematics, a transformation refers to a change in the position, shape, or size of a geometric figure. Transformations can be classified into four types: translation, rotation, reflection, and dilation.

Here,

The transformation of LM to L'M' involves both translation and reflection. To see the translation, we can compare the x- and y-coordinates of L and L', as well as M and M':

The x-coordinate of L' is 5 units more than the x-coordinate of L: -2 = -7 + 5.

The y-coordinate of L' is 2 units less than the y-coordinate of L: -4 = -2 - 2.

The x-coordinate of M' is 5 units more than the x-coordinate of M: 5 = 0 + 5.

The y-coordinate of M' is 2 units less than the y-coordinate of M: 3 = 5 - 2.

Therefore, we can conclude that LM is translated 5 units right and 2 units down to L'M'. This eliminates options OB and OD. To see the reflection, we can compare the x-coordinates of L and M, and their respective x-coordinates in L' and M':

The x-coordinate of L is negative and the x-coordinate of M is positive.

The x-coordinate of L' is negative and the x-coordinate of M' is positive.

Therefore, we can conclude that LM is reflected over the y-axis to L'M'. This eliminates option OA. Therefore, the correct answer is option OC.

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The value of missing side lengths are,

m = 4/√3

h = 2 / √3

We have to given that;

A triangle is shown in figure.

Now, We can formulate;

sin 60° = 2 / m

√3/2 = 2 / m

m = 4/√3

cos 60° = h / m

1/2 = h / (4/√3)

1/2 = √3h / 4

2 = √3h

h = 2 / √3

Thus, The value of missing side lengths are,

m = 4/√3

h = 2 / √3

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Answer:

6

Step-by-step explanation:

bc

The median is the accurate reflection of the data

501, 900, 971, 977, 1071

The mean would be

summation of the values

mean = 884

The median is the value that occurs in the middle = 971

The median would be more accurate reflection of the data. 501 is an outler. This is what caused us to have a mean that is c loser to 884. Hence we have 971 the median to be more accurate reflection of the data

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Answer: The man will make deposits for 66 - 45 = 21 years, or 21 x 4 = 84 quarters.

The quarterly interest rate is 4% / 4 = 1%.

Let's use the formula for the future value of an annuity:

FV = PMT x ((1 + r)^n - 1) / r

where FV is the future value of the annuity, PMT is the periodic payment, r is the interest rate per period, and n is the number of periods.

In this case, PMT = $2500, r = 1%, and n = 84. Substituting these values into the formula, we get:

FV = $2500 x ((1 + 0.01)^84 - 1) / 0.01

FV = $2500 x (5.409 - 1) / 0.01

FV = $2500 x 540.9

FV = $1,352,250

Therefore, there will be $1,352,250 in the account when the man retires at age 71.

The value of amount of R4 670,00 is due to be paid back after​ 1.47 years.

We have to given that;

A discounted loan of R4 670,00 at a simple discount rate of 14% per year is offered to Mr. Mothapo.

And, The actual amount of money that Mr. Mothapo receives is R3 852,75.

Now, Let n be the loan period.

Hence, We get;

4670,00 = 385275 (1 + 0.14)ⁿ

1.212121 = (1.14)ⁿ

n log 1.14 = log 1.2121

n = 1.47

Thus, The value of amount of R4 670,00 is due to be paid back after​ 1.47 years.

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The required maximum area of the rectangle is 17.5m

The perimeter of a rectangle is = 70m.

We have to find The maximum area of rectangle .

A rectangle will have the maximum possible area for a given perimeter when all the sides are the same length. Since every rectangle has four sides, if you know the perimeter, divide it by four.

Maximum area of rectangle is = 70 / 4

                                                   = 17.5

The required value of maximum area of the rectangle is 17.5m.

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the perimeter of a rectangle is 70m. What are the dimensions that will produce the maximum area of such a rectangle

The calculated values of the sum of interior angles of the shapes are 1080 degrees and 360 degrees

The formula of the sum of interior angles of a polygon is

Sum = (n − 2) * 180

For the octagon, we have

n = 8

This means that

Sum of angles = (8 − 2) * 180

Evaluate

Sum of angles = 1080

For the square, we have

n = 4

This means that

Sum of angles = (4 − 2) * 180

Evaluate

Sum of angles = 360

Hence, the sum of interior angles of the shapes are 1080 degrees and 360 degrees

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Answer:

45w + 180 = 900

45w = 720

w = 16 weeks

There is higher variability of y with large values of x.

There is a non linear relation between x and y.

We observe from figure b that for lower values of x the residuals are negative and as the values of x increases the residuals become positive but are distributed very close to the zero line.

Again for the extremely high values of x the residuals become negative following a decreasing non linear trend. Hence, from the residual plot we can conclude that the relation between x and y is a non linear one. Hence, the correct option is fourth option i.e, There is a non linear relation between x and y.

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The single transformation that maps shape P onto shape Q is given as follows:

Reflection over the line y = 1.

Examples of transformations are given as follows:

The orientation of the figure changed, hence it underwent a reflection.

The intercepts are given as follows:

y = 4 and y = -2.

The line of reflection is the mean of the coordinates of the intercepts, hence:

(4 - 2)/2 -> y = 1.

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Step-by-step explanation:

You would need a calculator in the degree function I believe, but basically for Number 1,

you would set it up in calculator as Sin ^-1 (6/9) or write it down as Sin(X)=(6/9)

2. would be Cos(45)= (X/4) meaning you'd do Cos(45) times 4.

3. is basically Tan(60) =(x/4) so it's basically Tan(60) times 4.

make sure your calculator is in degree mode ? Sorry, I don't remember if you need to be in radiant or degree mode. I can comment or make an edit when I remember.

The table is completed as follows:

A proportional relationship is a type of relationship between two quantities in which they maintain a constant ratio to each other.

The equation that defines the proportional relationship is given as follows:

y = kx.

In which k is the constant of proportionality, representing the increase in the output variable y when the constant variable x is increased by one.

From the first row, when x = 0.2, y = 2, hence the constant is given as follows:

k = y/x

k = 2/0.2

k = 10.

Hence the equation is:

y = 10x.

Then the numeric values are given as follows:

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The solutions to the equation 100+ (n - 2)² = 149 are n = 9 and n = -5

From the question, we have the following parameters that can be used in our computation:

The equation 100+ (n - 2)² = 149

Express as

100+ (n - 2)² = 149

Subtract 100 from both sides of the equation

So, we have the following representation

(n - 2)² = 49

Take the square root of both sides

n - 2 = 7 and n - 2 = -7

Add 2 to both sides

n = 9 and n = -5

Hence, the solutions to the equation 100+ (n - 2)² = 149 are n = 9 and n = -5

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Answer:

Perimeter of rhombus = 4(9.7) = 38.8 cm

Area of rhombus = 4(1/2)(6.2)(7.4)

= 91.76 cm^2