what is the base and coefficient of the exponential function:

Area of a triangle ABC with the given vertices is 3 square units.

Given that, the vertices of a triangle ABC, A(3,-6), B(5,-6), and C(7,–9).

In Geometry, a triangle is a three-sided polygon that has three edges and three vertices. The area of the triangle is the space covered by the triangle in a two-dimensional plane.

Area of a triangle = [tex]\frac{1}{2} ( |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|)[/tex]

Here, (x1, y1) = A(3,-6), (x2, y2) = B(5,-6), and (x3, y3) = C(7,–9)

Now, the area of a triangle = 1/2 (|3(-6+9)+5(-9+6)+7(-6+6)|)

= 1/2 (|3(3)+5(-3)+7(0)|)

= 1/2 (|(9-15)|)

= 1/2 × 6

= 3 square units

Therefore, area of a triangle ABC with the given vertices is 3 square units.

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The graph shows that the range of the function is 0.9 ≤ M ≤ ∞.

A linear function can be represented by a line. The standard form for this equation is: y=mx+b , for example, y=5x+8.

All functions present their domain and range. The domain of a function is the set of input values for which the function is real and defined. In the other words, when you define the domain, you are indicating for which values x the function is real and defined. While the domain is related to the values ​​of x, the range is related to the possible values ​​of y that the function can have.

From the graph it is possible to see that: the function is a linear function, the values of the coordinate x are represented by the volume (liters) while the values of the coordinate y are represented by the mass (kg).

The question asks for the range of the function. Therefore, you should indicate the possible values ​​of y that the function can have.

For this, you should analyze the axis-y. See that for x=0, the graph shows y =0.9. Therefore, the function starts for values of y >= 0.9 kg. It is possible to verify that when the volume increases, the mass also increases. With this information, you can find that the range is 0.9 ≤ M ≤ ∞.

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Let

x -----> pounds of one kind of nuts ($2.00 per pound)

y ----> pounds of other kind of nuts ($2.90 per pound)

we have that

2x+2.90y=2.50(16) ------> equation 1

x+y=16-----> x=16-y -----> equation 2

solve the system of equations

substitute equation 2 in equation 1

2(16-y)+2.90y=40

solve for y

32-2y+2.90y=40

2.90y-2y=40-32

0.90y=8

y=8.89

Find out the value of x

x=16-8.89

x=7.11

therefore

the answer is

Let the two numbers be represented with x and y.

Quotient of two numbers = -1:

Difference of two numbers = 8:

From the first equation, make x the the subject.

Thus, we have:

x = -y

Substitute -y for x in equation 2:

-y - y = 8

-2y = 8

Divide both sides by -2:

Now, substitute -4 for y in equation 2:

x - y = 8

x - (-4) = 8

x + 4 = 8

Subtract 4 from both sides:

x + 4 - 4 = 8 - 4

x = 4

Therefore,

x = 4 and y = -4

Thus, the numbers are 4 and -4

ANSWER:

4 and -4

here is your answer I hope this helps

Angles are given below.

When two straight lines or rays intersect at a single endpoint, an angle is created. The vertex of an angle is the location where two points come together. The Latin word "angulus ," which means "corner," is where the term "angle" originates. When a transversal connects two coplanar lines, alternate interior angles are created. They are located on the transverse sides of the parallel lines, but on the inner side of the parallel lines. At two different locations, the transversal passes through the two lines that are coplanar.

Given,

∠6 and ∠7 = Vertical angles

∠2 and ∠8 = Same side exterior angles

∠1 and ∠5 = Corresponding angles

∠3 and ∠6 = Adjacent angles

∠2 and ∠7 = Alternate exterior angles

∠4 and ∠6 = Same side interior angles

∠1 and ∠2 = Linear pair

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Answer:

C

Step-by-step explanation:

Angle 7 is congruent to angle 5 by the corresponding angles theorem, and angles 5 and 12 are supplementary because they are consecutive interior angles.

Thus, angles 7 and 12 are supplementary.

By the compound interest formula, you know that

Answer: P can be anything from 8 to below

Example: 8, 7, 6, 5, 4, 3, 2, 1, 0, -1 .....

We have the following equation:

And we want to identify values who satisfy the equation. And the possible answers for this case are:

x=0

Since :

Other two possible answers are:

Answer:

Step-by-step explanation:

Set up either long division or synthetic div.  I'd do the latter.

Your divisor for synth. div. should be -7.

                          __________  

Then              -7 /  2      9      k

                                   -14   +35

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

                            2      -5      k+35                 Rem is 32;

                                                                    Let 32 = k + 5 and solve for k:

                                                                          k = -3.

Let's check that.  Is k correct?

Then              -7 /  2      9      -3

                                   -14   +35

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

                            2      -5      -3+35  = 32

Since the rem is 32, we are correct; k = -3.  

The expression for the volume of the box is:

Mathematically, there is no restriction for the values of x, but phisically we know that x is a length and has a positive value, so x>0.

Also, we know that x can not be largest than half of the width, that is the smallest dimension of the piece of paper.

As the width is 11, we then know that x is smaller than 11/2=5.5.

In conclusion, the domain for x is:

The solutions are x=2 and x=7 approximately.

Because of our domain definition, we know that x=7 is not a valid solution, so the value of x that maximizes the volume is x=2.

The volume for x=0 is 0. Then, it will increase its value until x=2, where it reaches the maximum volume. From x=2 to x=5.5, the volume decrease until reaching v=0 at x=6.5.

Answer:

Domain: 0

Value of x that maximizes the volume: x=2.

From x=0 to x=2 the volume of the box increases.

The value of k is 9 for the function g.

To solve this problem we should have a brief concept of algebraic functions.

To solve this problem we have to follow a few steps.

Here f is a function of x and the relation with the function denotes as x²-2x. Also, g is a function of x and the relation with the function denotes as x− 2.

If we put, x = 2 on  f(x) = x²-2x. We can write, f(2) = 2²-2.2 = 4 - 4 = 0.

If we put, x = 2 on g (x) = x − 2. We can write, g(2) = x− 2= 2- 2 = 0.

Hence, we can conclude that f(2) = g(2) = 0.      ( proved)

Here, g(k) = 7. So, x = k in this relation.

We have to put x= k on g(x) = x− 2 ; now we can write, g (k) = k− 2.

g (k) = k− 2 = 7 as per the question. Therefore k = 7 + 2 = 9

The value of k is 9.

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The correct question is,

If f (x) = x²-2x

g (x) = x − 2

1) prove that : f(2) = g(2)

2) If g (K) = 7, find the value of k

amount of income tax:

Answer:

$7499.2

If Point m represents the opposite of negative 1/2 and point n represents the opposite of positive 5/2. Then M is 1/2 and N is -5/2.

A number system is defined as the representation of numbers by using digits or other symbols in a consistent manner.

A number line is a picture of a graduated straight line that serves as visual representation of the real numbers.

Given that point M represents the opposite of negative 1/2. Which means opposite of -1/2. The opposite of  -1/2 means positive of 1/2. The sign changes.

Point N represents the opposite of positive 5/2. Which means opposite of 5/2. The opposite of  5/2 means negative of 5/2. The sign changes.

opposite of positive 5/2 is -5/2.

Now let us plot this values on a number line. 1/2 is 0.5 and -5/2 means -2.5.

The graph is attached below.

Hence M is 1/2 and N is -5/2

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SOLUTION:

After the first year, the price of the laptop computer is;

After the second year, the price of the laptop computer is;

After the third year, the price of the laptop computer is;

After the fourth year, the price of the laptop computer is;

After the fifth year, the price of the laptop computer is;

After the sixth year, the price of the laptop computer is;

After the seventh year, the price of the laptop computer is;

CORRECT ANSWER: 7 years

Answer:

Diverge i think.

Step-by-step explanation:

See the photo

The area of the shaded region will be 28 cm². The shaded region is a combination of two rectangles.

The area of the shaded area is the difference between the total area of the polygon and the area of the portion of the polygon that is not shaded. In polygons, the area of the shaded component might appear in two different ways. A polygon's sides or its center are both potential locations for the shaded area.

From the triangle, it is obtained that the shaded region is the combination of a 4×6 rectangle and a 2×2 square.

Area of shaded region = Area of rectangle + Area of square

Area of shaded region = (6-2)×6 + 2×2

Area of shaded region  = 4×6 + 4

Area of shaded region = 24 + 4

Area of shaded region =  = 28 cm²

Thus, the area of the shaded region will be 28 cm². The shaded region is a combination of two rectangles.

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The inverse of the functions given as y = 5(x-2) and f^-1(x) = x²

A function's inverse function reverses the action of a function, or f.

a) From the given table, we can use the coordinate points (-5, 1) and (0, 2)

The standard linear equation is y = mx + b

m = 2-1/0-(-5)

m = 1/5

Since the y-intercept is (0, 2), hence the required function is y = 1/5 x + 2.

y = 1/5x + 2

x = 1/5 y + 2

1/5 y = x - 2

y = 5(x-2)

b) For the function g(x) = √x

y = √x

x = √y

y = x²

f^-1(x) = x²

This gives the inverse of the function.

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Step-by-step explanation:

so, I understand, the given series is written in binary form.

a1 = 1 = 1×2⁰ = 1

a2 = 11 = 1×2¹ + 1× 2⁰ = 3

a3 = 111 = 1×2² + 1×2¹ + 1×2⁰ = 7

a4 = 1111 = 1×2³ + 1×2² + 1×2¹ + 1×2⁰ = 15

a5 = 11111 = 1×2⁴ + 1×2³ + 1×2² + 1×2¹ + 1×2⁰ = 31

...

we see, that

an = 2×(an-1) + 1

a1 = 1

a2 = 2×a1 + 1

a3 = 2×a2 + 1 = 2×(2×a1 + 1) + 1 = 4×a1 + 2 + 1

a4 = 2×a3 + 1 = 2×(2×a2 + 1) = 2×(2×(2×a1 + 1) + 1) + 1 =

= 8×a1 + 2×2 + 2 + 1 = 8×a1 + 7

...

an = (2^(n-1))×a1 + an-1

because

an = 2×(an-1) + 1,

an-1 = (2^(n-1))×a1 - 1

therefore,

an = 2×(2^(n-1))×a1 - 1 = (2^n)×a1 - 1

the sequence of the sums of the first n elements

s1 = a1 = 1

s2 = a1 + a2 = 1 + 3 = 4

s3 = a1 + a2 + a3 = 7 + 3 + 1 = 11

s4 = a1 + a2 + a3 + a4 = 15 + 7 + 3 + 1 = 26

...

(i)

it is NOT a geometric sequence.

for a geometric sequence

an/an-1 = r, and r must be a constant ratio for any n.

but

7/3 = 2.333333...

15/7 = 2.142857143...

these are different, so, the sequence itself is not geometric.

neither is the sequence of the sums of the series. because

11/4 = 2.75

26/11 = 2.363636363...

are different.

1, 2, 4, 8, 16, 32, ... is a geometric sequence (constant r = 2).

but not

1, 3, 7, 15, 31, ...

(ii)

11 111 111 111 111 111 111 in base 2.

the utmost right position is the 2⁰ position. every position further to the left multiples the position value by 2. it is the same process as for numbers in base 10 (just there every position value is multiplied by 10).

we have 6×3 + 2×1 positions = 20 positions.

so, the position values go from 2⁰ to 2¹⁹.

as per the formula for "an" up there, we get

a20 = (2²⁰)×a1 - 1 = 1,048,576 - 1 = 1,048,575

We are asked to find the final average retail price of the vehicle after the given additions and deductions.

The average retail value is $15,857

Add $60 for heated outside mirrors.

Add $250 for rear and side airbags.

Add $175 for cruise control.

Add $100 for remote keyless entry.

Deduct $750 for excessive mileage.

Therefore, the average retail price is $15,692

We know that

• The width of the rectangle is 3.6 inches.

• The perimeter is 72 inches.

The perimeter formula for a rectangle is

Where P = 72, w = 3.6, and we have to solve for l.

We are to determine the equation of line by interpreting tabulated results between an independent variable ( x ) and a dependent variable ( y ).

A function is usually expressed as follows:

The above notation gives us the output ( y ) which is a function of input variable ( x ). This means that whatever relationship these two variables have the value of output ( y ) is related to the imput variable ( x ).

We are given a table/list of values of output ( y ) corresponding to each value of input variable ( x ) as follows:

Input ( x ) Output ( y )

3 -5

6 -4

9 -3

There are a series of steps that we must take to arrive at the equation that relates two variables.

Step 1: Determine the type of relationship between two variables by intuition

The first step in the process is the hardest of all. We have to critically analyze each input value ( x ) and its corresponding output value ( y ) with successive pair of values.

There are many types of relationships possible ( polynomial order, exponential, logarithmic, trigonometric, radical, etc .. ).

We can conjure up a way by comparing outputs of successive values to determine the type of relationship possible.

So looking at the first value:

The successive value:

The next successive value:

Here if scrutinize between each successive value of input variable ( x ) we see that there is a "3 unit step-up" in each pair of values i.e ( 3 -> 6 -> 9 ).

Next we compare each output values ( y ) for successive pairs. We see that with every step increase of 3 units in ( x ) value there is an increase of ( 1 ) unit in output value i.e ( -5 -> -4 -> -3 ).

Conclusion: Combing the result of above analysis we see that with each 3 step increase in input value ( x ) there is an increase in output value ( y ) by 1 unit.

This gives us the idea that the two variables are linearly related to one another.

Therefore, the type of relationship is:

Step 2: Recall the equation for the type of relationship between two vairbales x and y

Once we have determined the type of relationship between two variables. We will have to resort to our equation bank and pluck out the corresponding equation that expresses a LINEAR relationship i.e equation of a straight line.

The slope-intercept form of a straight line is:

Step 3: Determine the complete equation of function by defining the arbitrary constants.

The above equation is valid for all straight lines that express a linear relationship. However, we seek to find a unique straight line for the given set of points.

Every unique straight line equation would have either of the constants different. The constants defined in a striaght line equation are:

To determine these constants we will use the given pairs of coordinates of input and output variables, x and y respectively.

To determine the slope (m) of an equation:

The above expression relates the change in output value ( y ) with respect to change in input variable ( x ).

To determine the constant ( m ) we will use the conclusion from Step 1:

"3 step increase in input of ( x ) value there is an increase in output value ( y ) by 1 unit."

Therefore,

To determine the value of y-intercept ( c ). We will plug in the value of ( m ) into the general equation of a straight line written in step 2:

Now, we will use any pair of input and output value.

Substitute the pair of values into the derived equation expressed above and solve for constant ( c ):

Note: The above step implies that following equation must satisfy each and every data pair of point given to us ( table ). Or each and every value must lie on the line. For that each value must satisfy the equation of line.

Step 4: Write the complete equation of the relationship

Once we have evaluated the values of equation defining constants ( m and c ). We can simply plug in the values into the general equation relationship ( Linear - slope intercept form ) as follows:

Therefore, the equation for the set of values given to us is:

The given equation is

The slope of the given line is 4/3 because it's the coefficient of x.

Now, the new line has a slope of 4/3 too because parallel lines have equal slopes.

We know that the new line passes through the point (5, -8). Let's use the point-slope formula to find the equation.

Replacing the points and the slope, we have.

The values are as:

a) f(g(x)) = √(x+ 9)

b) (gf)(x)= √x +9

The core concept of mathematics' calculus is functions. The unique varieties of relations are the functions. In mathematics, a function is represented as a rule that produces a distinct result for each input x. In mathematics, a function is indicated by a mapping or transformation. Typically, these functions are identified by letters like f, g, and h. The collection of all the values that the function may input while it is defined is known as the domain. The whole set of values that the function's output can produce is referred to as the range. The set of values that might be a function's outputs is known as the co-domain.

Given:

f(x)=√x, g(x)=x+9

a) (fg)(x)=

f(g(x)) = f( x+9)

         = √(x+ 9)

Now, domain is all the input values

i.e., x=2, 4, 7

f(g(2)) = √(2+ 9)

          = √11

and, f(g(4)) = √(4+ 9)

          = √13

and, f(g(7)) = √(7 + 9)

          = √16

          = 4

b) (gf)(x)= g(f(x))

            = g(√x)

            = √x +9

Now, domain is all the input values

i.e., x=2, 4, 7

f(g(2)) = √2+ 9

and, f(g(4)) = √(4+ 9)

          = 2+9

          = 11

and, f(g(7)) = √7 + 9

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We are given the following expression:

We are also given that k is equal to 2/3. So, we can substitute that into the expression:

Due to order of operations, we have to do the division first, and then do the subtraction. To do division with fractions, we keep the first fraction the same and take the reciprocal of the second fraction. Then, we can multiply the two fractions. Let's do that:

Now, we can do the subtraction:

Therefore, our answer is 73/9

Using the probability concept, the odds of rolling a number greater than 5 is 1 :5

What is probability ?

probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur or how likely it is that a proposition is true . the probability of an event is a number between 0 and 1 , where ,roughly speaking ,0 indicates impossibility of the event and 1 indicates certainty.

The odd of a particular experiment is defined thus :

Number of possible outcomes greater than 5 : number of possible below or equal to 5

Sample space = {1, 2, 3, 4, 5, 6}

Outcomes greater than 5 = {6} = 1

Outcomes below or equal to 5 = {1, 2, 3, 4, 5} = 5

The odds equals to 1 : 5

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To simplify the expressions you have to combine the like terms.

This means that you'll solve the operations between the terms that have the same variables, for example x + 2x=3x

Or the terms that have no variables and are only numbers, for example 4+5=9

a. The expression is

Both terms have the same variable "x", so you can add them together. To do so, add the coefficients, i.e. the numbers that are being multiplied by x

And you get that the simplified expression is 7x

b. The expression is

In this expression you have two types of terms, the x-related terms and one constant. In this case you have to solve the operation for the x-related terms together and leave the constant as it is

The simplified expression is 8x-1

x is the ladder ,x =36

y is the distance between ladder and wall y=12

z is the wall

we have in this triangle , only the hypotenuse(x) and the adjacent side

so,'

Answer

Option B is correct.

Angle Y = 68º

Explanation

Similar triangles have the same set of angles in them.

All the corresponding angles are equal to each other.

So, if triangle ABC is similar to triangle XYZ

Angle A = Angle X

Angle B = Angle Y

Angle C = Angle Z

The order in which they are named determines the angles that are corresponding to each other.

So, if Angle B = 68º

Angle Y = Angle B = 68º

Hope this Helps!!!