12. Two functions are given below, one in equation form and the other in table form. Using these functions, give
the value of g (f (4)). Show how you arrived at your answer.
-2
f(x) -1
0
3
4
12
7
9
19
6
g(x)=3.25x+6

Answer:

Step-by-step explanation:

Ordered pairs

x = 1                        Given

y = 1 - 6 = - 5          Put the given into the equation

Ordered pair: (1,-5)

x = 4

y = 4 - 6

y = - 2

Ordered pair:(4,-2)

Graph

The reduced or lowest form of 15 / 40 is 3 / 8. The given statement is true.

First, let us understand about the reducing a fraction:

To express a fraction in the lowest terms, divide the numerator and denominator by the greatest common factor of the numerator and denominator.

We are given:

15 / 40 = 3 / 8

Let the LHS; 15 / 40

We need to write the following fraction in the lowest terms:

15 / 40

To write 15 / 40 in the lowest terms, we divide the numerator and the denominator by the greatest common factor of 15 and 40.

Since the factors of 15 are 1, 2, 3, 5, and 15 and the factors of 40 are 1, 2, 4, 5, 8, 10, 20, and 40, the greatest common factor of 15 and 40 will be 5.

So we will divide the numerator and the denominator of 15 / 40 by 5.

15 ÷ 5 = 3

40 ÷ 5 = 8

So, the reduced form of 15 / 40 is 3 / 8 which is 15 / 40 = 3 / 8.

Thus, the reduced or lowest form of 15 / 40 is 3 / 8. The given statement is true.

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The graph of the function f(x) = 1/4x + 3 is attached below and the point (-x, f(-4) ) is pointed on it.

Graph:

Graph refers the visual representation of the function of set of data.

Given,

Here we have the function f(x) = 1/4x + 3

Now, we have to plot the function into the graph and we also plot the point (-4, f(-4)) on it.

By using the graphing calculator, we have plot the function f(x) = 1/4x + 3, then we get the graph like the following.

In order to point the (-4, f(-4)),

We have to find the value of f(-4), by apply the value of x as -4 in the given function then we get the values of the function as,

=> f(-4) = 1/4 (-4) + 3

=> f(-4) = -1 + 3

=> f(-4) = 2

Therefore, the point is (-4, 2).

Now we have to plot this point on the graph, and the point is on the line of the given function.

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Answer:

[tex]y = 4(1.4)^x[/tex]

[tex]\text{at x = 5, }y=21.513[/tex]

Step-by-step explanation:

If you think about an exponential function in the form: [tex]f(x) = a(b)^x[/tex]

you'll realize a pattern, which is by definition.

Evaluating the function at x = 1: [tex]f(1) = a(b)^1 \implies a(b)[/tex]

Evaluating the function at x = 2: [tex]f(2) = a(b)^2 \implies a(b * b)[/tex]

Evaluating the function at x = 3: [tex]f(3) = a(b)^3\implies a(b * b * b)[/tex]

Each term just has an additional "b" that it's being multiplied by. This is by definition of an exponential function, repeated multiplication.

You'll also notice something else interesting, we can find this "b" value by dividing any term by it's previous term.

Take for example f(1) and f(2): [tex]\frac{a(b* b)}{a(b)}\to b[/tex]

As mentioned before, as x increases by one, we just have another term to multiply by. So if we divide by the previous term (where x is one less), then we should just have this "b" value, which is the generally expressed in either a growth or decay rate.

So now that we know that, we can divide f(2) by f(1) using the values shown on the graph:

given:

[tex]f(1) = 5.6\\f(2) = 7.84[/tex]

Let's divide the f(2) by f(1) to get: [tex]\frac{7.84}{5.6} \to1.4[/tex]

Lastly to find our "a" value, we just need to find the y-intercept, since when x = 0, the b will be equal to one (since anything raised to the power of zero is one), we can just look at the graph to see the y-intercept.

Looking at the graph, we can see the y-intercept is four, so a = 4.

Another way to do this algebraically rather than visually, would be to use our knowledge of exponentials to realize that as x increases by one (when we're going right), the y-value is just being multiplied by "b" as mentioned before.

So as x decreases by one (when we're going left) the y-value is just being divided by "b".

So f(0) should be equal to f(1) / b, and we know both values! Which are going to substitute into: [tex]\frac{5.6}{1.4}\to 4[/tex]

Anyways, once we plug in "a" and "b" into the standard form of an exponential function, we get: [tex]y = 4(1.4)^x[/tex]

We can now use this equation to find y-values that are not shown in the graph.

To find the x-value at x = 5, simply substitute in "5" as x

[tex]y = 4(1.4)^x\\\\\text{substitute 5 as x}\\\\y=4(1.4)^5\\\\y=4(5.37824)\\\\y\approx 21.513[/tex]

The composition of the functions, when evaluated in x = 4, is:

g( f(4)) =  39

Here we have two functions, one is a linear function:

g(x) = 3.25*x + 6

And the other is f(x), which is defined on a table.

We want to find the composition:

g( f(4)) = 3.25*f(4) + 6

Then we need to find the value of f(4), on the table we have the pair:

x   f(x)

4    12

Which means that:

f(4) = 12

Replacing that we get:

g( f(4)) = 3.25*f(4) + 6

g( f(4)) = 3.25*12 + 6 = 39

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Answer:

-46

Step-by-step explanation:

i hopethis helps you!!

Answer:

B. 4|−25| = $100

Step-by-step explanation:

Hope this helps!

The object had an initial velocity of 0m/s

Projectile motion is a form of motion experienced by an object or particle that is projected in a gravitational field, such as from Earth's surface, and moves along a curved path under the action of gravity only.

A projectile is an object upon which the only force acting is gravity. There are a variety of examples of projectiles. An object dropped from rest is a projectile (provided that the influence of air resistance is negligible). An object that is thrown vertically upward is also a projectile (provided that the influence of air resistance is negligible). And an object which is thrown upward at an angle to the horizontal is also a projectile (provided that the influence of air resistance is negligible). A projectile is any object that once projected or dropped continues in motion by its own inertia and is influenced only by the downward force of gravity.

Using equation of motion, we can find it's initial velocity.

v² = u² - 2gh

substituting the values and solving for u

But we don't know h; we can use the acceleration to find that

a = v / t

a = 18/4

a = 4.5 m/s²

v = u + at

18 = u + 4.5 * 4

u = 0

The initial velocity was 0m/s

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Caroline have $7901.5 more in her account than Qasim when they both invest together compounded for 10 years.

Compound interest is when an amount receives interest on top of it each time interest is paid on the original amount. The principal (original) sum and the interest that has already accrued over the course of previous periods are used to calculate compound interest.

here,

Caroline invested $80,000 in an account paying an interest rate of 8 1/8% compounded daily for 10 years,

A=P(1+r/n)^n/t

=$180,266.48

Caroline invested $80,000 in an account paying an interest rate of 8 1/8% compounded daily for 10 years,

A=P(1+r/n)^n/t

=$172,365.02

The difference between amounts,

=$180,266.48-$172,365.02

=$7,901.46

When they both invest which is compounded for 10 years, Caroline has $7901.5 more in her account than Qasim.

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Answer: 7901

Step-by-step explanation:

The area of the rectangle is 35.07 square feet which can be rounded to 35.1 square feet.

Given that:

Here ABDC is a rectangle and AD is diagonal. Hence, Δ ABC is a right triangle with AD = 9 feet and ∠CAD = 60°.

In ΔCAD,

sin ∠CAD = sin 60°= Perpendicular/Hypotenuse

sin 60° = AC/AD

√3/2 = AC/9

=> AC = 9√3/2 ft

cos ∠CAD = cos 60° = Base/ Hypotenuse

cos 60° = CD/9

1/2 = CD/9

=> CD = 4.5 ft

So, calculating the area of the rectangle = length x breadth

                                                                    = AC x CD

                                                                    = 9√3/2 x 9/2

                                                                     = 81√3/4 ft²

The area of the rectangle = 35.07 ft²  which can be rounded to 35.1 square feet.

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Using the linear function equation that models the situation given, the amount of credit left after 24 minutes of calls is: $25.2.

One of the ways to write a linear function equation is to express it in slope-intercept form as y = mx + b, where the slope is represented as m and the y-intercept is represented as b.

From the information given, we can deduce the following:

y = credit card remaining in dollars

x = minutes

Slope (m) = -0.13

A point = (x, y) = (32, 24.16)

Substitute m = -0.13, x = 32, and y = 24.16 into y = mx + b to find the value of b:

24.16 = -0.13(32) + b

24.16 = -4.16 + b

24.16 + 4.16 = b

28.32 = b

b = 28.32

Write the equation of the linear function by substituting b = 28.32 and m = -0.13 into y = mx + b:

y = -0.13x + 28.32

To find how much credit there was after 24 minutes of calls, substitute x = 24 into y = -0.13x + 28.32:

y = -0.13(24) + 28.32

y = 25.2

$25.2 credit was left.

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The statement that best describes how to determine if the plane clears the tower is;

Use the Pythagoras Theorem where the distance to the tower is a leg of the right triangle and the distance in the air is the hypotenuse. Find the other leg. Convert to feet to compare to the tower height

Given,

The length the plane will travel before lifting = 1500 yards

The distance further the plane will travel after lifting off the ground = 603 yards

The horizontal distance from the point of lifting off the ground to control tower = 600 yards

The horizontal distance from the point of liftoff to the control tower, the height of the control tower, and the distance of the plane's path after takeoff create a right triangle with the following sides:

The distance of the path of the plane after lifting off the ground = The hypotenuse side of the triangle

The horizontal distance from the point of lifting off the ground to control tower and the height of the control tower = The two legs of the right triangle

Let h stand for the height of the control tower, x for the horizontal distance from the plane's liftoff point to the control tower, and R for the length of the plane's route after liftoff, and we obtain;

h = √(R² - x²)

We have;

R = 603 yards

x = 600 yards

∴ h = √(R² - x²) = h = √(603² - 600²) = 60.07 yards

The height of the control tower, h = 60.07 yards

1 yard = 3 feet

∴ 60.07 yards = 3 × 60.07 feet ≈ 180.22 feet

Therefore, given that the height of the control tower = 198 feet, the plane at the height of approximately 180.22 feet clears the tower.

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The initial height of the rocket is 264 feet

From the given graph

The height of the rocket is h is feet and time taken t is in seconds

The graph is defined as the pictorial representation of the mathematical equation or function.  

The x axis of the graphs represents the Time in seconds

The y axis of the graph represents the height of the rocket h in feet

From the graph we can say that

At t = 0, the value of h = 264 feet

That means the height of the rocket is 264 feet when the time t = 0

Hence, the initial height of the rocket is 264 feet

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Answer:

-1/3

Step-by-step explanation:

Substituting into the slope formula, the slope is [tex]\frac{-6-(-5)}{-6-(-9)}=-\frac{1}{3}[/tex]

A baker makes 90 cakes in a 60-hour week

A baker makes 18 cakes in a 12-hour day.

So, the number of cakes she makes in an hour would be,

18/12 = 1.5 cakes/hour

We need to find the number of cakes she make in a 60-hour week.

Using unitary method we find the required number of cakes,

1.5 * 60 = 90

Therefore, a baker makes 90 cakes in a 60-hour week

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The probability that the student selected is 5/8.

According to the question,

We have the following information:

A class consists of 30 women and 18 men.

Now, total students in the class = Number of women + number of men

Total students = 30+18

Total students = 48

Now, the probability that the student is a woman:

Probability of an event = Number of women/total number of students

30/48

(More to know: probability of an event can not be greater than 1.)

Now, converting this into the simplest form by dividing both the numerator and denominator by 6:

5/8

Hence, the probability that the student selected is 5/8.

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According to the given temperature function, the substance will not be in liquid state in range is written as the inequality -30.874 < x < 566.121.  

Function:

Function refers the special relationship where each input has a single output. It is often written as "f(x)" where x is the input value.

Given,

Matter in a liquid state when it’s temperature is between it’s melting point and boiling point. Suppose that some substance has a melting point of -34.93 degrees Celsius and a boiling point of -332.29 degrees Celsius.

Here we need to find the range of temperatures in degrees Fahrenheit for which this substance is not in liquid state.

Here we have the following details

Melting point = -34.93C°

Boiling point = -332.29C°

Function = C = 5/9 (F - 32)

n order to find the inequality of the function we have to apply the value of  boiling and melting point in it,

First, we have to apply the value of melting point, then we get,

=> -34.93 = 5/9 (F - 32)   Distribute

=> -34. 93 = 5/9 F - 160/9    Multiply both sides by 9

=> -314.37 = 5F - 160   Add 160 on both sides

=> -154.37 = 5F     Divide both sides by 5

=> -30.874= F

Therefore, the melting point in F= -30.874.

Similarly, for the boiling point it can be calculated as,

=> -332.29 = 5/9 (F - 32)   Distribute

=> -332.29 = 5/9 F - 160/9    Multiply both sides by 9

=> -2990.61 = 5F - 160   Add 160 on both sides

=> -2830.61 = 5F     Divide both sides by 5

=> -566.121= F

Therefore, the boiling point in F= -566.121

So, the resulting inequality is -30.874 < x < 566.121.

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The value of car depreciates by $310 per month.

What is Depreciation?

The reduction in the value of an asset due to wear and tear is termed as Depreciation.

Solution:

Value of Car after 7 months = $26,930

Value of Car after 30 months = $19,800

So, it can be said that in 23 months the value of car depreciated by $7,130.

Since, straight line depreciation method is been followed,

Therefore, monthly depreciation will be calculated by dividing $7,130 to 23

= $7,130/23

= $310

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From the calculation, the two fractional parts are; 6/14.

We know that a fraction is composed of a numerator and a denominator. A numerator is the number that is found at the top while the denominator is the number that is found below. We are told that we have the 6/7 and we are told to express it as the sum of two equal fractional parts;

Hence let the fractional parts be x. We know that the fractional parts are equal hence;

2x = 6/7

x = 6/7 * 1/2

x = 6/14

To check our working;

6/14 + 6/14 = 12/14 = 6/7

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Answer:

4/7

Step-by-step explanation:

(8-4)/(10-3) = 4/7

parallel = same slope

Answer:

The slope of line n is 4/7

Step-by-step explanation:

First find the slope of line m

m = ( y2-y1)/(x2-x1)

m = (8 - 4)/(10 -3)

m = 4/7

Parallel lines have the same slope

The slope of line n is the same as slope of line m

The slope of line n is 4/7

The equation of the line in slope-intercept form is y = 2x-8.

According to the question,

We have the following information:

Slope of the line = 2

Points through which the line is passing = (3,-2)

We know that the following formula is used to find the equation of the line passing through a point:

y-y' = m(x-x') where m is the slope of the line

In this case, we have the followings values:

m = 2

x' = 3 and y' =-2

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

y+2 = 2x-6

Subtracting 2 from both sides of the equation:

y = 2x-6-2

y = 2x-8

Hence, the equation of the line in slope-intercept form is y = 2x-8.

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The trigonometric function is useful to find the sides of a right-angle triangle. The value of x using the trigonometric function sin will be 7.884.

The fundamental 6 functions of trigonometry have a range of numbers as their result and a domain input value that is the angle of a right triangle.

The trigonometric function is only valid for the right angle triangle and it is 6 functions which are given as sin cos tan cosec sec cot.

Given right angle triangle,

Use sin trigonometric function.

Sin21° = x=22

x = 22 × 0.358

x = 7.884

Hence "The trigonometric function is useful to find the sides of a right-angle triangle. The value of x using the trigonometric function sin will be 7.884".

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For the given angles , the measure of the missing angles are as follow:

a. m ∠POQ = 36° , m ∠POR = 62°

b. m ∠RST = 40°

As given in the question,

For the given angles , the measure of the missing angles are given as follow:

a. Q is in the interior of angle POR ,

Measure of angle POR = 2x -2

Measure of angle POQ = x + 4

Measure of QOR = 26°

From the given relation of the angles we have,

m ∠POQ + m ∠QOR = m ∠POR

⇒ x + 4 + 26° = 2x -2

⇒2x -x = (2+ 4+ 26)°

⇒ x= 32°

m ∠POQ = x + 4

               = 32° + 4°

               = 36°

m ∠POR = 2x -2

              = 2(32) -2

              = 62°

b. Q bisects angle RST

m ∠RSQ = 8x -12

m ∠RST = 10x

From the given relation we have,

8x -12 = (1/2)10x

⇒ 8x -12 = 5x

⇒8x -5x = 12

⇒3x =12

⇒ x= 4

m ∠RST = 10x

             = 10(4)

             = 40°

Therefore, for the given angles , the measure of the missing angles are as follow:

a. m ∠POQ = 36° , m ∠POR = 62°

b. m ∠RST = 40°

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The length of the border around the cake is the same as the perimeter of the cake, which is D. 44 inches.

The perimeter is the sum of the lengths of the four sides of the rectangular cake.

In other words, the perimeter is the total lengths of the boundary.  The result of the perimeter is always a linear measure in units.

We can compute the perimeter of a rectangle by adding up its four dimensions as follows:

Length = 13 inches

Width = 9 inches

Perimeter = 2(L + W)

= 2(13 + 9)

= 44 inches

Thus, this cake has Option D as the length of the border around it.

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The slope of the line that passes through the points (-1, 6) and (14, 3) is - 1 / 5.

We know that the slope of a line is given by the formula,

m = (y₂ - y₁) / (x₂ - x₁)

where (x₂, y₂) and (x₁, y₁) are the coordinates of any point on the line of slope.

We are given the following points;

(-1, 6) and (14, 3)

substitute the given value in the slope formula, we will get;

m = (3 - 6) / (14 - (- 1))

m = - 3 / 14 + 1

m = - 3 / 15

m = - 1 / 5

So, the slope is - 1 / 5.

Thus, the slope of the line that passes through the points (-1, 6) and (14, 3) is - 1 / 5.

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The measure of  the angle AEG = 92°

Corresponding angles are the angles that are formed when two parallel lines are intersected by the transversal. The opening and shutting of a lunchbox, solving a Rubik's cube, and never-ending parallel railway tracks are a few everyday examples of corresponding angles. These are formed in the matching corners or corresponding corners with the transversal.

∠CFH + ∠EFC = 180( angles on a straight line)

making ∠EFC subject and substituting ∠CFH = 2x + 20, we have

∠EFC = 180 - ( 2x + 20)

∠EFC = 180 - 2x -20

∠EFC = 160 - 2x

but ∠EFC = ∠BEF ( alternate angles are equal)

substituting their values

160 - 2x = 3x - 10

collect like terms

-2x - 3x = -10 - 160

-5x = -170

x = -170/-5

x = 34

Also ∠EFC = ∠AEG = 160 -2x ( corresponding angles are equal)

∠AEG = 160 -2X, but x = 34

∠AEG = 160 - 2(34)

∠AEG =160 - 68 = 92°

In conclusion, The value of the ∠AEG = 92°

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