Evaluate the function at the indicated values if possible. If an indicated value is not in the​ domain, say so.
f left parenthesis x right parenthesis equals StartFraction x plus 7 Over x squared minus 9 EndFraction
​; f left parenthesis negative 7 right parenthesis​, f left parenthesis 2 right parenthesis​, f left parenthesis 3 right parenthesis

Answer:

Proof is in the explanation.

Step-by-step explanation:

I'm going to use mathematical induction.

That means we are going to show:

1) For n=3 the expression given is a multiple of 6. (We started at n=3 because it says n>2.)

2) If the base cases check out, then we are going to assume (n-2)(n-1)(2n-3) is a multiple of 6, then show ([n+1]-2)([n+1]-1)(2[n+1]-3) is also a multiple of 6.

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

Proof:

Base case (n=3):

(3-2)(3-1)(2*3-3)

1(2)(6-3)

2(3)

6

6 is a multiple of 6 since 6(1)=6.

After the base case (for all natural numbers greater than 2):

Assume there is integer k such that:

6k=(n-2)(n-1)(2n-3).

We are going to show 6m=([n+1]-2)([n+1]-1)(2[n+1]-3) where m is a integer.

([n+1]-2)([n+1]-1)(2[n+1]-3)

(n-1)(n)(2n-1)

(n-2+1)(n)(2n-1)

(n-2)(n)(2n-1)+1(n)(2n-1)

(n)(n-2)(2n-1)+1(n)(2n-1)

(n-1+1)(n-2)(2n-1)+1(n)(2n-1)

(n-1)(n-2)(2n-1)+1(n-2)(2n-1)+1(n)(2n-1)

(2n-1)(n-2)(n-1)+1(n-2)(2n-1)+1(n)(2n-1)

(2n-3+2)(n-2)(n-1)+1(n-2)(2n-1)+1(n)(2n-1)

(2n-3)(n-2)(n-1)+2(n-2)(n-1)+1(n-2)(2n-1)+1(n)(2n-1)

6k+2(n-2)(n-1)+1(n-2)(2n-1)+1(n)(2n-1)

6k+2(n^2-3n+2)+1(2n^2-5n+2)+2n^2-n

6k+6n^2-12n+6

6(k+n^2-2n+1)

where k+n^2-2n+1 since integers are closed under addition and multiplication (referring to the n^2, the n*n part).

Since we have found an integer m, k+n^2-2n+1, such that

6m=([n+1]-2)([n+1]-1)(2[n+1]-3)

then we have shown for all integers greater than 2 we have that

(n-2)(n-1)(2n-3) is divisible by 6.

//

Answer:  The expressions for the length and width of the new patio are

[tex]\ell=x+4,~~w=x-4.[/tex]

And the area of the new patio is 384 sq. feet.

Step-by-step explanation:  Given that Stephen has a square brick patio. He wants to reduce the width by 4 feet and increase the length by 4 feet.

The length of one side of the square patio is represented by x.

We are to write the expressions for the length and width of the new patio and then to find the area of the new patio if the original patio measures 20 feet by 20 feet.

Since Stephen wants to reduce width of the patio by 4 feet, so the width of the new patio will be

[tex]w=(x-4)~\textup{feet}.[/tex]

The length of the patio is increased by 4 feet, so the length of the new patio will be

[tex]\ell=(x+4)~\textup{feet}.[/tex]

Now, if the original patio measures 20 feet by 20 feet, then we must have

[tex]w=x-4=20-4=16~\textup{feet}[/tex]

and

[tex]\ell=x+4=20+4=24~\textup{feet}.[/tex]

Therefore, the area of the new patio is given by

[tex]A_n=\ell \times w=24\times16=384~\textup{sq. feet}.[/tex]

Thus, the expressions for the length and width of the new patio are

[tex]\ell=x+4,~~w=x-4.[/tex]

And the area of the new patio is 384 sq. feet.

Final answer:

Stephon is altering his square patio's dimensions by reducing the width by 4 feet and increasing the length by 4 feet. For an original side length of 20 feet, the new dimensions are 24 feet by 16 feet, resulting in an area of 384 square feet.

Explanation:

Stephon has a square brick patio and is planning on changing its dimensions. Initially, the patio is a square with each side measuring x feet. To find the expressions for the new length and width of the patio after the alterations:

Given that the original side length of the patio is 20 feet, we can substitute this value into the expressions:

To find the area of the new patio, multiply the new length by the new width:

Area = Length × Width = 24 feet × 16 feet = 384 square feet.

To isolate the variable terms on one side and the constant terms on the other side of the equation 3x - 5 = -2 + 10, add 2 to both sides, simplify to 3x - 3 = 10, then add 3 to both sides to get the final simplified equation 3x = 13.

The equation 3x - 5 = -2 + 10 needs to be rearranged to isolate the variable terms on one side and the constant terms on the other. To do this, follow these steps:

Now, we have successfully isolated the variable terms (3x) on one side of the equation and the constant terms (13) on the other side.

Answer:

Step-by-step explanation:

note :

the area of a circle  is : A= π×r²    ....r : radius and : D=2r   ... D : diameter

47) D = 10 cm      r= 10cm/2 = 5 cm

A=π×r²  = 3.1448×5²  =78.62cm²

48) A= 3.1448×2²= 12.58cm²

Answer:

800 milliliters

Step-by-step explanation:

we know that

To find out the total amount Kelly drank, add the amount of coffee and the amount of yogurt and convert the result to milliliters.

so

0.5+0.3=0.8 liters

Remember that

1 liter= 1,000 milliliters

so

0.8 liters=0.8*1,000=800 milliliters

Answer:

800 mL

Step-by-step explanation:

Because we know that

1 liter equals 1000 milliliters

So 0.5+0.3=0.8

0.8 Liters=0.8*1,000 ML

Answer: Hello there!

here we have two equations:

1) x + 3y = 7

2) 2x + 4y = 8

a) first we want to isolate x in the first equation:

x + 3y = 7

x = 7 -3y

done!

b) now we want to replace it in the second equation, and in this way get a equation that depends only on the variable y.

2x + 4y = 8

2(7 - 3y) + 4y = 8

c) now we sole this equation and obtain the value of y.

14 - 6y + 4y = 8

14 - 2y = 8

-2y = 8 - 14 = -6

y = 6/2 = 3

d) now we have the value of y, and we can substitute it on the equation that we got in the part a)

x = 7 - 3y

x = 7 - 3*3 = 7 - 9 = -2

e) now we knowt that x = -2 and y = 3, then the pair (x,y) can be written as:

(-2,3).

The solution to the system of equations, as an ordered pair, is (-2,3).

System of linear equations is the given term math for two or more equations with the same variables. The solution of these equations represents the point at which the lines intersect.

The question gives step by step of the solution for the system of linear equations. The exercise found the value of y, then you should find the value of x.

The step 4 of the question shows: x = 7 − 3(3). Therefore:

x = 7 − 3(3)

x = 7 − 9

x= -2

You can check the values found for x and y from equation 2x + 4y = 8. Therefore:

2*(-2)+4*(3)

-4+12=8

Thus, the values found for x and y are correct.

Learn more about the system of equations here:

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[tex]\bf a^3b^{-2}c^{-1}d\implies \cfrac{a^3d}{b^2c}\qquad \begin{cases} a=2\\ b=4\\ c=10\\ d=15 \end{cases}\implies \cfrac{2^3\cdot 15}{4^2\cdot 10}\implies \cfrac{120}{160}\implies \cfrac{3}{4}[/tex]

Answer:

z = 5/3 or 1 2/3

Step-by-step explanation:

2/3·z=10/9

Multiply each side by 3/2

3/2*2/3·z=10/9*3/2

z = 30/18

We can simplify by dividing the top and bottom by 6

z = 5/3

Changing to a mixed number

z =1 2/3

Answer:

1⅔ [OR 5⁄3]

Step-by-step explanation:

2 × ? = 10

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

3 × ? = 9

That would be 1⅔.

I am joyous to assist you anytime.

Answer:

1. Dispersion: how data is distributed

2. Inter quartile range:the difference between the largest and smallest of the middle 50% of the data set

3. Lower Quartile:the median of the lower half of the data set; a value which 25% of the data set falls below

4. Outlier:a data value that is far from the others

5. Percentile: a value below which a certain percentage of the data set falls; the median is the 50th percentile.

6. Range:the difference between the largest and smallest of the numbers in a set

7. Upper Quartile:the median of the upper half of the data set; a value which 75% of the data set falls below

Final answer:

In statistics, 'dispersion' refers to the distribution of data, the 'inter-quartile range' is the spread of the middle 50% of data, 'lower quartile' (Q1) is the value below which 25% of data falls, an 'outlier' is a data point far from the others, 'percentile' is a value below a certain percentage of data, 'range' is the difference between the largest and smallest data values, and 'upper quartile' (Q3) is the value below which 75% of the data falls.

Explanation:

To correctly match the terms to their definitions from the provided options:

Dispersion is matched to 'how data is distributed.'

Inter-quartile range (IQR) is 'the difference between the largest and smallest of the middle 50% of the data set.'

Lower quartile (also known as the first quartile or Q1) is 'the median of the lower half of the data set; a value which 25% of the data set falls below.'

Outlier is 'a data value that is far from the others.'

Percentile is 'a value below which a certain percentage of the data set falls; the median is the 50th percentile.'

Range is 'the difference between the largest and smallest of the numbers in a set.'

Upper quartile (also known as the third quartile or Q3) is 'the median of the upper half of the data set; a value which 75% of the data set falls below.'

Answer:

45.55 m to the nearest hundredth.

Step-by-step explanation:

tan 26 = opposite / adjacent side = h / 90   where h = height of the dam - Scarlett's height.

The height of the dam =

y = 90 tan 26

= 43.895 m

Now we need to add Scarlett's height  

= 45.55 m.

Answer:

grg

Step-by-step explanation:

rgg

Answer:

it must be 9

Step-by-step explanation:

it is a simple division.

Answer:

4 units

Step-by-step explanation:

just took a quiz and got it right

Answer: 4

Step-by-step explanation:

took the test

Answer:

29.1 cm

Step-by-step explanation:

Circumference of a circle is:

C = 2πr

Given that C = 183 cm:

183 = 2πr

r = 183 / (2π)

r ≈ 29.1 cm

Answer:

1.584

Step-by-step explanation:

1.584 = 3.96 × 0.4

Answer:

3.96*0.4 is 1.584

Answer:

=$2739.81

Step-by-step explanation:

To find the total amount if the interest is compounded, we use the compound interest formula.

A=P(1+R/100)ⁿ

A is the amount, P- principal, is the invested amount R is the % interest rate, n is the number if periods.

If compounded semi-annually, it means we have two periods in 1 year

The rate is also divided by 2

Thus 25 years have (25×2) = 50 periods.

A= 750(1+5.25/200)⁵⁰

=750(1.02625)⁵⁰

=$2739.81

Answer:

D. Negative slope

Step-by-step explanation:

Since it is a proportional relation, it must be linear. A line has a slope. Since it is decreasing, it is a negative slope.

Answer:B 42

Step-by-step explanation:

V=pi/4 x D^2 x H

3.14/4 x 36 x 1.5 = 42 ft^3

Answer:

B) 42

Step-by-step explanation:

The area of a cylinder is pi*r^2*h where r is the radius and h is the height. Plugging in 3 for the radius(radius=diameter/2) and 1.5 for height you get area=pi*9*1.5. Assuming pi as 3.14 you get 3.14*9*1.5 which is 42.39 which is approx. 42.

Answer:

This is the definition of rational number.

These include: Integers, Terminating Decimals, Repeating Decimals, or Proper, Improper, and Mixed Fractions where each part is an integer.  

Step-by-step explanation:

This is the definition of rational number.

Here are some examples of rational number:

-3  (negative integers are included because they can be rewritten; here -3=-3/1)

5 (positive integers are included because they can be rewritten; here 5=5/1)

0 (neutral integers are included also because 0/4 or 0/1 are still 0)

5/3 (impropert fractions where top and bottom are integers; this is already written in the form required)

1   2/3  (mixed fractions because they can rewritten as improper fractions with top and bottom as integers; example here this 5/3)

2/5  (proper fractions where top and bottom are integers; this is already written in the form required)

.55555555555...=[tex]. \overline{5}[/tex] (repeating decimals; example this one can be written as 5/9)

.23 (terminating decimals; example this can be written as 23/100 )

The restriction on the domain of the function [tex]f(x) = {x + 5}/{x - 2}.[/tex] is that x cannot be equal to 2, since it would make the denominator zero, which is undefined in real numbers.

The student is asking to identify the restrictions on the domain of the function [tex]f(x) = {x + 5}/{x - 2}.[/tex] The domain of a function includes all the values that x can take for which the function is defined. In the case of a rational function, any values that make the denominator zero must be excluded from the domain since division by zero is undefined.

In this function, the denominator is x - 2. Therefore, the value that makes the denominator zero is x = 2. To identify the restrictions on the domain of [tex]f(x) = {x + 5}/{x - 2}.[/tex] we set the denominator equal to zero and solve for x:

Hence, the only restriction on the domain of this function is that x cannot be 2. So the domain of f(x) is all real numbers except x = 2.

Answer:

There are 97920 ways to formed the committee

Step-by-step explanation:

* Lets solve explain the combination

- combination is a collection of the objects where the order doesn't

 matter

- Combinations is nCr, where n is the total number and r is the number  

 of the choices

# Example: chose a group of three students from the group of 10

  students  n = 10 and r = 3,then 10C3 is 120

* Lets solve the problem

- The club has 30 members

- There are 3 lawyers, 4 teachers , 5 doctors in the group

- We want to formed a committee of 8 contains 1 teacher, 2 lawyers,

  2 doctors

∵ There are 4 teachers, we want to chose 1 of them

∴ 4C1 = 4

∵ There are 3 lawyers, we want to chose 2 of them

∴ 3C2 = 3

∵ There are 5 doctors, we want to chose 2 of them

∴ 5C2 = 10

- To find how many ways multiply 4C1 by 3C2 by 5C2

∵ 4C1 × 3C2 × 5C2 = 4 × 3 × 10 = 120

∵ The total numbers of the teachers, the lawyers and the doctors is

   4 + 3 + 5 = 12 members from the 30 members

∴ There are 120 ways to chose 5 members from 12 members

∵ The committee has 8 members

∴ We want to chose another 3 from the rest of the members

∵ The rest of the members = 30 - 12 = 18

∴ We must to find 18C3

∵ 18C3 = 816

- To find the total ways of the 8 members multiply the ways of the 5

  members and the 3 members

∴ The total number of ways = 120 × 816 = 97920

∴ There are 97920 ways to formed the committee

Answer:

x + 5y = 7

Step-by-step explanation:

You can find which one is correct by plugging in -3 for each x value and then solving for y.

x + 5y = 7

(-3) + 5y = 7

5y = 10

y = 2

When x = -3, y=2

(2) + 5y = 7

5y = 5

y = 1

When x=2, y=1

The equation of straight line that represents a line that passes through points (−3, 2) and (2, 1) is Option (D) x + 5y = 7.

The equation of a straight line passing through the points (x1,y1) and (x2,y2) and having slope m is given by -

y - y1 = m(x - x1) .

The slope m, can be calculated as m =  (y2 - y1)/(x2 - x1) .

Thus the equation of straight line in slope intercept form is -

[tex]y - y1 = \frac{y2 - y1}{x2 - x1} (x - x1)[/tex]

Given points are (-3,2) and (2,1) .

We have x1 = -3 , x2 = 2 , y1 = 2 , y2 = 1

Slope (m) = (1 - 2)/(2 - (-3)) = -1/5

Putting the required values to find the equation of straight line in slope intercept form -

⇒ y - 2 = (-1/5)*(x - (-3))

⇒ (y - 2)*5 = -1*(x + 3)

⇒ 5y - 10  =  -x - 3

∴  x  +  5y = 7

The required equation is Option (D) x + 5y = 7.

Thus, the equation of straight line that represents a line that passes through points (−3, 2) and (2, 1) is Option(D) x + 5y = 7.

To learn more about equation of straight line in slope intercept form , refer -

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Answer:

y = - 6x + 5

Step-by-step explanation:

The equation of a line in slope- intercept form is

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

y = [tex]\frac{1}{6}[/tex] x + 3 ← is in slope- intercept form

with slope m = [tex]\frac{1}{6}[/tex]

Given a line with slope m then the slope of a line perpendicular to it is

[tex]m_{perpendicular}[/tex] = - [tex]\frac{1}{m}[/tex] = - [tex]\frac{1}{\frac{1}{6} }[/tex] = - 6, hence

y = - 6x + c ← is the partial equation of the perpendicular line.

To find c substitute (- 3, 23) into the partial equation

23 = 18 + c ⇒ c = 23 - 18 = 5

y = - 6x + 5 ← equation of perpendicular line

Answer:

3x+10

Step-by-step explanation:

( - x + 12) - (  - 4x + 2)

Distribute the minus sign

( - x + 12) + 4x - 2

Combine like terms

3x +10

[tex]\bf sin(y)+cos(y)+tan(y)sin(y)=sec(y)+cos(y)tan(y) \\\\[-0.35em] ~\dotfill\\\\ sin(y)+cos(y)+tan(y)sin(y)\implies sin(y)+cos(y)+\cfrac{sin(y)}{cos(y)}\cdot sin(y) \\\\\\ sin(y)+cos(y)+\cfrac{sin^2(y)}{cos(y)}\implies \stackrel{\textit{using the LCD of cos(y)}}{\cfrac{sin(y)cos(y)+cos^2(y)+sin^2(y)}{cos(y)}} \\\\\\ \cfrac{sin(y)cos(y)+\stackrel{cos^2(y)+sin^2(y)}{1}}{cos(y)}\implies \cfrac{sin(y)cos(y)+1}{cos(y)} \\\\\\ \cfrac{sin(y)}{cos(y)}\cdot cos(y)+\cfrac{1}{cos(y)}\implies tan(y)cos(y)+sec(y)[/tex]

Answer:

A.2sinxcosx

Step-by-step explanation:

We know the trig identity

Sin (2a) = 2 sin a cos a

sin (2x) = 2 sin x cos x

Answer:

2sinxcosx

Step-by-step explanation:

A P E X

Answer:

270

Step-by-step explanation:

base area=18

18*15=270

The volume of the triangular prism is equal to [tex]270[/tex] cu. units.

" Volume is defined as the total space occupied by a three-dimensional object."

Formula used

Volume of a triangular prism = Area of the base × height

Area of the base [tex]= \frac{1}{2} \times base \times height[/tex]

According to the question,

Given dimensions,

Base of triangle [tex]= 9 units[/tex]

Height of the triangle [tex]=4 units[/tex]

Height of the triangular prism [tex]= 15 units[/tex]

Substitute the value in the formula to get the area of the base we have,

Area of the base [tex]= \frac{1}{2}\times 9\times 4[/tex]

                             [tex]= 18 square units[/tex]

Volume of a triangular prism [tex]= 18 \times 15[/tex]

                                                  [tex]= 270 cu.units[/tex]

Hence, the volume of the triangular prism is equal to [tex]270[/tex] cu. units.

Learn more about volume here

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To find the inverse of a function switch the place of y (aka f(x) ) with x. Then solve for y.

Original equation:

y = 2x + 17

Switched:

x = 2y + 17

Solve for y by isolating it:

x - 17 = 2y + 17 - 17

x - 17 = 2y

(x - 17)/2 = 2y/2

[tex]\frac{1}{2}x-\frac{17}{2}= y[/tex]

Hope this helped!

~Just a girl in love with Shawn Mendes

Answer:

[tex]\huge \boxed{y=-2}[/tex]

Step-by-step explanation:

First thing you do is switch sides.

[tex]\displaystyle 2y-3=-7[/tex]

Then add 3 from both sides of equation.

[tex]\displaystyle 2y-3+3=-7+3[/tex]

Simplify.

[tex]\displaystyle 2y=-4[/tex]

Divide by 2 from both sides of equation.

[tex]\displaystyle \frac{2y}{2}=\frac{-4}{2}[/tex]

Simplify, to find the answer.

[tex]\displaystyle -4\div2=-2[/tex]

[tex]\large \boxed{y=-2}[/tex], which is our answer.

Hope this helps!

−7=2y−3

Step 1: Flip the equation.

2y−3=−7

Step 2: Add 3 to both sides.

2y−3+3=−7+3

2y=−4

Step 3: Divide both sides by 2.

2y/2=-4/2

y=−2

Answer:

[tex]5 {x}^{2} + 9x - 1[/tex]

Step-by-step explanation:

A quadratic expresion is of the form

[tex]a {x}^{2} + bx + c[/tex]

where

[tex]a \ne0[/tex]

The give options are:

[tex]9x-2[/tex]

[tex]5{x}^{2}+9x - 1[/tex]

[tex]-2x^3+8 {x}^{2} -7x +1[/tex]

[tex]x^4-12x^3+8 {x}^{2} -7x +1[/tex]

From the given options, the only expression which is quadratic is

[tex]5 {x}^{2} + 9x - 1[/tex]

where a =5, b=9 and c=-1

Therefore the correct choice is the second option.

The quadratic expression is [tex]\(5x^2 + 9x - 1\),[/tex]  as it has the highest degree of 2 among the given options. :

A quadratic expression is a polynomial of degree 2, which means the highest power of [tex]\(x\)[/tex]  in the expression is  [tex]\(x^2\).[/tex]

Let's examine the given expressions:

1. [tex]\(9x - 2\)[/tex]  - This is a linear expression (degree 1).

2. [tex]\(5x^2 + 9x - 1\)[/tex]  - This is a quadratic expression (degree 2).

3. [tex]\(-2x^3 + 8x^2 - 7x + 1\)[/tex]  - This is a cubic expression (degree 3).

4. [tex]\(x^4 - 12x3 + 8x^2 - 7x + 1\)[/tex]  - This is a quartic expression (degree 4).

Therefore, the expression that represents a quadratic expression is:

[tex]\[5x^2 + 9x - 1\][/tex]

So, the correct choice is [tex]\(5x^2 + 9x - 1\).[/tex]