Plz help with the problem in the picture. Thanks

Let f be a function defined on all of that satisfies the additive condition is,

f(x) = [tex]kx[/tex] for every x∈ R.

It is given that:

f: R → R is a continuous function such that:

[tex]f(x+y)[/tex] = [tex]f(x) + f(y)[/tex]    ∀  x , y ∈ R  (Equation-1)

Now, let us assume f(1)=k

Also,

f(0) = 0( Since, f(0)=f(0+0)

That is,

f(0)=f(0)+f(0)

By using property (1)

Also,

f(0)=2f(0)

That is,

2f(0)-f(0)=0

That is,

f(0)=0  )

Also,

f(2)=f(1+1)

That is,

f(2)=f(1)+f(1)         ( By using property (1) )

i.e.

f(2)=2f(1)

That is,

f(2)=2k

Similarly for any m ∈ N

f(m)=f(1+1+1+...+1)

That is,

f(m)=f(1)+f(1)+f(1)+ ....... +f(1) (m times)

That is,

f(m)=mf(1)

That is,

f(m) = [tex]mk[/tex]

Now,

f(1) = [tex]f(\frac{1}{n} +\frac{1}{n} +......\frac{1}{n} )[/tex]

f(1) = [tex]f(\frac{1}{n} )+f(\frac{1}{n} )+......f(\frac{1}{n} )[/tex]

f(1) = n [tex]f(\frac{1}{n} )[/tex]

f(1) = k

[tex]f(\frac{1}{n} )[/tex] = k * [tex]\frac{1}{n}[/tex]

Also,

when x∈ Q

[tex]x = \frac{p}{q}[/tex]

Then,

[tex]f(\frac{p}{q})[/tex] = [tex]f(\frac{1}{q} )+ f(\frac{1}{q} )+ .....f(\frac{1}{q} )[/tex]

[tex]f(\frac{p}{q})[/tex] = p * [tex]f(\frac{1}{q} )[/tex]

[tex]f(\frac{p}{q})[/tex] = [tex]p\frac{k}{q}[/tex]

[tex]f(\frac{p}{q})[/tex] = [tex]k\frac{p}{q}[/tex]

So,

[tex]f(x) = kx[/tex] for all x belongs to Q

Now, as we know that:

Q is dense in R.

So Э x∈ Q' such that Э a seq < [tex]x_{n}[/tex] >  belonging to Q such that:

< [tex]x_{n}[/tex] > ⇒[tex]x[/tex]

Now, we know that: Q'=R

This means that:

Э α ∈ R

such that Э sequence [tex]a_{n}[/tex] such that:

[tex]a_{n}[/tex] belongs to Q,

[tex]a_{n}[/tex] ⇒ [tex]\alpha[/tex]

[tex]f(a_{n} )= ka_{n}[/tex]

( since [tex]a_{n}[/tex] belongs to Q )

Let f is continuous at x=α

This means that:

[tex]f(a_{n} )[/tex] ⇒ [tex]f(\alpha )[/tex]

k [tex]a_{n}[/tex] ⇒ [tex]f(\alpha )[/tex]

k [tex]a_{n}[/tex] ⇒ k [tex]\alpha[/tex]

This means that:

[tex]f(\alpha )[/tex] = k [tex]\alpha[/tex]

This means that:

f(x) = [tex]kx[/tex] for every x∈ R

Therefore,

Let f be a function defined on all of that satisfies the additive condition is,  f(x) = [tex]kx[/tex] for every x∈ R.

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Answer: 3/8

Step-by-step explanation:

5/8 of them want to, so you subtract 5/8 from 1

1-(5/8)

(8/8)-(5/8)

((8-5)/8)

3/8

The value of x and y in right angle triangle is  16√2 and 16

What is a Right Angle Triangle ?

A right triangle is a triangle having one right angle or two perpendicular sides. It is also referred to as a right-angled triangle, right-perpendicular triangle, orthogonal triangle, or previously rectangled triangle. The relationship between the sides and various angles of the right triangle serves as the basis for trigonometry.

The Pythagorean theorem, often known as Pythagoras' theorem, in Euclidean geometry relates the three sides of a right triangle. This statement states that the area of the square whose side is the hypotenuse is equal to the sum of the areas of the squares on the other two sides.

In the triangle 2 angles are 90° and 45° respectively.

The third angle is  = 180°+90°-45°=45°

The triangle is issosceles triangle

So the side which is not x is 16.

Now by using Pythagoras Theorem

x = √16^2+16^2 = 16√2

The value of x is 16√2

The value of y is y^2 + 16^2 = 16√2^2

y^2 = 512 - 256

y^2 = 256

y = 16

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The angles of the triangle KHP are P ≈ 118.555°, H ≈ 34.614°, K ≈ 26.844°, respectively.

Here we have the case of a triangle where the lengths of three sides are known and measures of angles must be determined. The missing angles can be found by means of law of cosine, whose equations related to the triangle are introduced below;

p² = h² + k² - 2 · h · k · cos P

h² = p² + k² - 2 · p · k · cos H

k² = h² + p² - 2 · h · p · cos K

Given;

h, k, p - Sides

H, K, P - Angles

If we know that h = 16.1, p = 24.9 and k = 12.8, then the measures of the angles are;

cos P = (p² - h² - k²) / (- 2 · h · k)

cos P = (24.9² - 16.1² - 12.8²) / (- 2 · 16.1 · 12.8)

cos P = - 0.478

P ≈ 118.555°

cos H = (h² - p² - k²) / (- 2 · p · k)

cos H = (16.1² - 24.9² - 12.8²) / (- 2 · 24.9 · 12.8)

cos H = 0.823

H ≈ 34.614°

cos K = (k² - h² - p²) / (- 2 · h · p)

cos K = (12.8² - 16.1² - 24.9²) / (- 2 · 16.1 · 24.9)

cos K = 0.892

K ≈ 26.844°

Triangle KHP has the following missing angles: P ≈ 118.555°, H ≈ 34.614°, K ≈ 26.844°;

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Answer: k*6 and then you find the reciprocal of k*6 which is 1 over k*6

Step-by-step explanation:

Answer:

[tex] \displaystyle{ {f}^{ - 1} (x) = {(x - 4)}^{3} + 1}[/tex]

Step-by-step explanation:

Let y = f(x)

[tex] \displaystyle{y = \sqrt[3]{x - 1} + 4}[/tex]

Finding an inverse, swap x and y

[tex] \displaystyle{x = \sqrt[3]{y- 1} + 4}[/tex]

Then solve for y, first subtract 4 both sides

[tex] \displaystyle{ x - 4= \sqrt[3]{y- 1} + 4 - 4} \\ \\ \displaystyle{ x - 4= \sqrt[3]{y- 1} }[/tex]

Cube both sides

[tex] \displaystyle{ {(x - 4)}^{3} = \left(\sqrt[3]{y- 1} \right)^{3} } \\ \\ \displaystyle{ {(x - 4)}^{3} = y - 1}[/tex]

Add both sides by 1

[tex] \displaystyle{ {(x - 4)}^{3} + 1 = y - 1 + 1} \\ \\ \displaystyle{ {(x - 4)}^{3} + 1 = y}[/tex]

The final y is the inverse. Hence:

[tex] \displaystyle{ {f}^{ - 1} (x) = {(x - 4)}^{3} + 1}[/tex]

Answer:

Out of all possible choices you've shown, I'd say the best answer would be A

Step-by-step explanation:

[tex]{ \qquad\qquad\huge\underline{{\sf Answer}}} [/tex]

[tex]\qquad \sf  \dashrightarrow \: 816 = 600( 1 + 9r)[/tex]

[tex]\qquad \sf  \dashrightarrow \: 816 = 600 + 5400r[/tex]

[tex]\qquad \sf  \dashrightarrow \: 5400r = 816 - 600[/tex]

[tex]\qquad \sf  \dashrightarrow \: 5400r = 216[/tex]

[tex]\qquad \sf  \dashrightarrow \: r = \dfrac{216}{5400} [/tex]

[tex]\qquad \sf  \dashrightarrow \: r= \dfrac{4}{100} [/tex]

[tex]\qquad \sf  \dashrightarrow \: r = 0.04[/tex]

The equation, 816 = 600(1 + 9r), represents the amount of money earned on a simple interest savings account. On solving the linear equation for r, we get r = 0.04 (a).

The linear equations in one variable is an equation which is expressed in the form of ax+b = 0, where a and b are two integers, and x is a variable and has only one solution.

Given in the question,

[tex]816 = 600(1 + 9r)\\\\\frac{816}{600} = 1+9r\\ \\\frac{816}{600} -1 = 9r\\\\\frac{216}{600} = 9r\\\\r = \frac{216}{600* 9 } = 0.04[/tex]

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a) The total cost of purchasing the car that is worth $32,000 with the harmonized sales tax (HST) of 13% is $36,160.

b) By leasing the car instead of purchasing it outright, the customer will save $14,760.

c) The lease options for the lessee after returning the car include:

A lease is a financing option for the use of capital assets.

With a lease, the lessee uses the asset for a specific period and the lessor receives periodic payments.

There are operating and finance (capital) leases.  Though, under new accounting standards, all leases are to be classified as finance leases.

Value of the car = $32,000

Harmonized Sales Tax (HST) = 13%

Cost of the car with HST = $36,160 ($32,000 x 1.13)

Down payment = $1,000

Financing part = $35,160 ($36,160 - $1,000)

Installment payments = 48

Periodic payments = $425

Total lease cost = $21,400 ($1,000 + 48 x $425)

Savings (advantages) from leasing = $14,760 ($36,160 - $21,400)

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We decide on 320 as the best-predicted value. EV(node 4) = 320. The answer is option (a).

What is Probability?

Simply put, the probability is the likelihood that something will occur. When we don't know how an event will turn out, we might discuss the likelihood or likelihood of several outcomes. Statistics is the study of occurrences that follow a probability distribution.

Probability of S1 = 0.6

Probability of F = 0.40

Value of node y =?

As per the given value if probability of S1 = 0.6

Then the probability of S2 = 0.4

Using payoff values in table equation from except value.

[tex]EV(d_{i})=[/tex] ∑ [tex]V_{ij} S_{J}[/tex]

where [tex]V_{ij}[/tex] = value of payoff

We compute the expected value for each of decisions

EV(Node 8) = 0.6*100+0.4*300

                  = 180

EV(Node 9) = 0.6*400+0.4*200

             = 320

For Node 4, We select the best value from node 8 and node9

The best decision attractive is the line complex branch that  EV(node 8) = 180, For node 9,

Therefore, We select the best-predicted value i.e 320.

EV(node 4) = 320

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The common difference of the sequence is -1/2

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

3/8, -1/8, -5/8, -9/8

The common difference is the difference between successive terms

So, we have

Difference = -1/8 - 3/8

Evaluate the difference

Difference = -1/2

Hence, the common difference is -1/2

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Answer:

m=5000 members

Step-by-step explanation:

15000-2500=12500

12500/25=500

Answer:

18

Step-by-step explanation:

One trip around the gym would be 138 ft.

If we cut the diagonal, we use the Pythagorean theorem to find the distance of the hypotenuse, which is 51 ft.

One trip with the shortcut is 119 ft. (24 + 45 + 51 = 119).

Subtract 119 ft. from 138 feet and you have 18 ft. shorter route than walking the entire perimeter.

a. To subtract like terms, subtract the coefficients, not the variables b. Like terms are terms that contain the same variable e. Negative (negative x) is equivalent to Negative x, so Negative 5 x minus x = negative 6 x.

The expression can be made simpler by using these facts.

We may take the coefficients of the similar terms, which are x and -x, out using fact a. In this instance, the expression is simplified to read as Negative 5 x minus x = Negative 6 x.

According to fact b, terms that have the same variable raised to the same power are referred to as being like terms. As we can see, there are two like terms—Negative y and -2y—in this situation that can be joined. The similar terms' coefficients, which are -1 and -2, can be added. The total is y-3.

Negative (negative x) is equal to negative x, according to Fact e. We can see that Negative 5 x is Negative (negative x) in this situation and can be reduced to Negative 6 x.

Therefore, Negative 6 x -3 y + 3 z would be the abbreviated expression.

Due to the fact that fact C's computation does not use the same simplified formulation as the other facts, it is important to note that fact C is incorrect.

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The likelihood that the basketball player will convert at least six of his or her eight free throw tries is 0.32.

The simple definition of probability is the likelihood that something will occur. We can discuss the probabilities of different outcomes, or how likely they are, whenever we are unsure of how an event will turn out. Statistics refers to the study of events subject to probability.

The ratio of all potential cases to the number of cases that are favorable to an event determines its likelihood when there is no reason for us to believe that any one of these circumstances will happen more frequently than any other, making them all equally likely.

The likelihood that an event will occur is quantified by probability. The probability of an event is the relative frequency throughout time of that event. Probabilities are all numbers between 0 and 1, inclusive—that is, any numbers between 0 and 1.

Therefore, the correct answer is option d) 0.32.

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Analyzing the coordinates being provided, the scale factor is found to be 0.5, hence the image of point A(12,4) under the same dilation and the center of dilation as origin is A'(6,2).

A dilation is a transformation in Euclidean space that enlarges or reduces all the points in a figure by a scale factor. In the example given, the dilation maps (6, 10) to (3, 5) with a scale factor of 0.5, and the center of dilation is the origin (0,0). The image of a point A(x,y) under this dilation would be A'(x*0.5, y*0.5). It means the point is scaled down by 0.5 in both x and y coordinate.

The dilation described above can be used in real-world applications such as image processing, where it can be used to reduce the size of an image without losing important details, or to enlarge an image without introducing distortion. It can also be used in architectural design to create smaller or larger versions of floor plans and blueprints. Additionally, it can be used in cartography to change the scale of maps without altering the shape of the features on the map.

Analyzing the given dilation map , mapped from (6,10) to (3,5) ,the center of the dilation being origin. We can see that the scaling factor is 0.5.

So , using the same dilation for point A(12,4), The image of A will be,

A'(12*0.5,4*0.5) = A'(6,2)

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There are 192 pennies in a stack that is 1 ft high.

A measurement unit is a standard quality used to express a physical quantity. Also it refers to the comparison between the unknown quantity with the known quantity.

Given that;

Stack is 1 feet high.

Now, We know that;

⇒ 1 inch = 16 pennies

And, 1 feet = 12 inches

So, We get;

⇒ 1 ft = 12 inches

         = 12 × 16 pennies

         = 192 pennies

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The answer is $10.4

Simple interest is an interest charge that borrowers pay lenders for a loan. It is calculated using the principal only and does not include compounding interest. Simple interest relates not just to certain loans. It's also the type of interest that banks pay customers on their savings accounts.

Given here: Makayla invested $130. She earned a simple interest of 4% per year on the initial investment.

Then SI = 130×4×2 /100

              =$10.4

Hence, Simple Interest recieved is $10.4

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Any triangle's three interior angles add up to 180 degrees, according to the Triangle Sum Theorem.

Any triangle's three interior angles add up to 180 degrees, according to the Triangle Sum Theorem. m∠1+m∠2+m∠3=180∘.

Here,

According to the triangle sum theorem, a triangle's total angles equal 180°. Therefore, using the Substitution property

=> (3x)° + 90° + (x+10)° = 180°,

=>  m(A,B,C) = 180.

To find x, combine similar terms to get

=> 4x + 100 = 180. Then, using the subtraction condition of equality, convert => 13 1/2 to 80.

=> Finally, divide 80 by  5 ( 13 1/2) to find x.

The substitution property may be used to get the measure of the angle A, yielding mA = 5 ( 13 1/2) °.

When the expression is finally simplified, m < A = 66 2/3.

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Answer:

------------------------------------

The pedestal is the combination of two rectangular prisms.

One of them has dimensions:

Its volume is:

The other one has dimensions:

Its volume is:

Total volume is:

Answer:

112 in³

Step-by-step explanation:

The pedestal is made up of two rectangular prisms.

[tex]\boxed{\begin{minipage}{5.8 cm}\underline{Volume of a rectangular prism}\\\\$V=w\cdot l\cdot h$\\\\where:\\ \phantom{ww}$\bullet$ $w$ is the width of the base. \\ \phantom{ww}$\bullet$ $l$ is the length of the base.\\\phantom{ww}$\bullet$ $h$ is the height of the prism.\\\end{minipage}}[/tex]

The dimensions of the largest rectangular prism are:

Therefore, the volume of the largest rectangular prism is:

[tex]\begin{aligned}\implies V&=2 \cdot 9 \cdot 4\\&=18 \cdot 4\\&=72\;\rm in^3\end{aligned}[/tex]

The dimensions of the smaller rectangular prism are:

Therefore, the volume of the smaller rectangular prism is:

[tex]\begin{aligned}\implies V&=2 \cdot 5 \cdot 4\\&=10\cdot 4\\&=40\;\rm in^3\end{aligned}[/tex]

The volume of the pedestal is the sum of the volumes of the largest and smaller rectangular prisms:

[tex]\implies V=72+40=112\; \rm in^3[/tex]

Therefore, the volume of the pedestal is 112 in³.

The lines are perpendicular

What is slope-intercept?

The slope-intercept form of a linear equation is a way of writing the equation of a line in the form y = mx + b, where m is the slope of the line and b is the y-intercept (the point where the line crosses the y-axis).

the equation of a line in slope-intercept form is

y = mx + c ( m is the slope and c the y- intercept )

3x + 5y = 10 ( subtract 3x from both sides )

5y = - 3x + 10 ( divide through by 5 )

y = -  x + 2 ← in slope- intercept form

with slope m = -

---------------------------

5x - 3y = - 6 ( subtract 5x from both sides )

- 3y = - 5x - 6 ( divide through by - 3 )

y =  x + 2 ← in slope-intercept form

with slope m =

---------------------------

• Parallel lines have equal slopes

clearly, the lines are not parallel

• the product of the slopes of perpendicular lines equals - 1

-  ×  = - 1

thus the 2 lines are perpendicular to each other

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Step-by-step explanation:

a. 21/100

b.31/100

c. 4/25

d. 3/25

Answer:

a). t = 5 years.

b). n = 25000 words.

Step-by-step explanation:

From the question it have been given a formula

which is n = 2450t + 500

a) The year

n = 12750 words

n = 2450t + 500

= 12750 = 2450t + 500

= 12750 - 500 = 2450t

= 12250 = 2450t

= 12250/2450 = 2450t/2450.

5 years = t

Or

t = 5 years.

b). The words

t = 10 years

= n = 2450t + 500

= n = 2450(10) + 500

= n = 24500 + 500

n = 25000 words.

You need to know the direction of the line and at least one point on it in order to parametrize a line. x is  5 + 4.0t ,y is  0 + 4.0t

x = 5 + 4.0t

y = 0 + 4.0t

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His altitude at the halfway point is -50 feet

Step-by-step explanation:

At the start of his hike, his altitude is -120 feet.

His altitude had dropped 70 feet.

His altitude at the halfway point of his hike is

= -120 + 70 = -50

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Answer:

15/487 or 0.0308008= 0.03

Step-by-step explanation:

You must solve to isolate the x variable.

If the bumper is at -1.6 units and It is programmed to move -2.3units. The bumper should be placed at -3.9 units

The location of the bumper is calculated by the concept of addition that takes place when the two values has negative signs values.

The negative numbers are located by the left of zero on the number line. The two points are both negative and have the value of -1.6 units and -2.3 units

Summing the two units together will result to

= -1.6 units + (-2.3 units)

= -1.6 - 2.3

= -3.9 units

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Answer:

(a) D y = 6x + 7

(b) A 6

Step-by-step explanation:

6x -6 = -7   Subtract 6x from both sides

6x - 6x - y = -6x - 7

-y = -6x - 7    Divide all the way through by -1

y = 6x + 7

The slope is the number before the variable x. (6)

Answer:

Expressions B and C

Step-by-step explanation:

A: 20^2 - 18^2 = 400 - 324 = 76 (not A)

B: 8(4^2) + 2^4 = 128 + 16 = 144 (which is equivalent to 12^2)

C: 15^2 - 3^4 = 225 - 81 = 144 (which is equivalent to 12^2)

Hope this helps!