One way to find the inverse of a function is by first swapping x and y, then solving for y, like this:
[tex]\begin{gathered} y=(x-4)^2+2\text{ }\Rightarrow x=(y-4)^2+2 \\ \end{gathered}[/tex]Now, let's solve for y, like this:
[tex]\begin{gathered} x=(y-4)^2+2 \\ x-2=(y-4)^2+2-2 \\ (y-4)^2=x-2 \\ \sqrt[]{\mleft(y-4\mright)^2}=\sqrt[]{x-2} \\ y-4=\sqrt[]{x-2} \\ y-4+4=\sqrt[]{x-2}+4 \\ y=\sqrt[]{x-2}+4 \end{gathered}[/tex]Then, the inverse function of y = (x-4)^2+2 is:
[tex]y=\sqrt[]{x-2}+4[/tex]B and Care sets of real numbers defined as follows.
Answer:
[tex]\begin{gathered} B\cap C=\phi \\ (-\infty,\text{ 1)}\cup\lbrack9,\infty) \end{gathered}[/tex]Step-by-step explanation:
Solve this situation with the help of the number line, if B and C are sets of real numbers defined as follow:
The intersection is an interval that lies within all of the given intervals. If no such intersection exists then the set is empty.
In this case, for the intersection between B and C:
[tex]\begin{gathered} B\cap C=\phi \\ \end{gathered}[/tex]For the union between B and C:
[tex](-\infty,\text{ 1)}\cup\lbrack9,\infty)[/tex]Which of the following is a solution to the inequality below?
Answer:
4u + 6 > 30
4u > 24
u > 6
Solution is u = 6
Answer:
4u+6>30
4u>30-6
4u>24
u>6
Answer:
u>6
Toni decides to plant a 2-foot wide rectangular flower garden along one side of the pool and patio but outside the fence. She measures the length of the fence to be 44 feet long. What is the area of the flower garden?
If she decides to plant a 2-foot wide rectangular flower garden along one side of the pool and patio but outside the fence. She measures the length of the fence to be 44 feet long. The area of the flower garden is 88 square feet.
Area of the flower gardenUsing this formula to determine the area of the flower garden
Area = Width × Length
Where:
Width = 2 feet
Length = 44 feet
Let plug in the formula
Area = 2 × 44
Area = 88 square feet
Therefore the area is 88 square feet.
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Help I’m stuck ‼️‼️‼️ Hw due in a couple minutes
The lines AD and BC cross at a point where we have two pairs of vertically opposite angles.
The angles labelled (2x +50) and 100 are vertically opposite angles.
Vertically opposite angles are equal. Therefore;
[tex]\begin{gathered} 2x+50=100 \\ \text{Subtract 50 from both sides} \\ 2x+50-50=100-50 \\ 2x=50 \\ \text{Divide both sides by 2} \\ \frac{2x}{2}=\frac{50}{2} \\ x=25 \end{gathered}[/tex]ANSWER:
The value of x is 25. The correct answer is option A
Given the following table of values determine the value of X where f(x) has a local minimum. Assume that f is continuous and differentiable for all reals
We have to find the value of x for which f(x) has a minimum.
Extreme values of f(x), like minimum or maximum values, correspond to values of its derivative equal to 0.
In this case f'(x) = 0 for x = -2 and x = 0.
We can find if this extreme value is a minimum if the second derivative f''(x) is greater than 0.
In this case, f'(x) = 0 and f''(x) > 1 for x = 0.
Then, x = 0 is a local minimum.
Answer: x = 0
Which of the following sampling methods would most likely have the smallest margin of erro?A. Roll a die 1000 times and estimate the proportion of 5's that result.OB. Sample 250 registered voters in a large city and ask them their political preference and use the results to estOC. Flip a coin 100 times and estimate the proportion of "heads" that resul.OD. Sample 10 adults and ask them if they support the current President's foreign policy and use this data to reReset SelectionMext
The sample methodology whose accuracy is better than another is the one with more approximation, this comes from the number of repetitions.
Therefore, option A is the one with more approximation, which mean the least error margin.
How4 x 8 sheet ofmanyply wood do you need tocover a 24 x 24 deck?
Given
Dimensions of deck = 24 by 24
dimensions of ply wood = 4 by 8
Find
Number of sheets of ply wood needed to cover the deck
Explanation
number of sheets = area of deck divided by area of 1 ply wood
so ,
area of deck =
[tex]\begin{gathered} 24\times24 \\ 576 \end{gathered}[/tex]and
area of ply wood =
[tex]\begin{gathered} 4\times8 \\ 32 \end{gathered}[/tex]so ,
number of sheets needed =
[tex]\begin{gathered} \frac{576}{32} \\ \\ 18 \end{gathered}[/tex]Final Answer
Hence , the required number of sheets of ply wood is 18
We are reviewing a module and I don't remember how to do it.
The coordinate of point X which is 5/6 of the distance between P and Q is 5/11
Here, we want to calculate the coordinates of point X which is 5/6 of the distance between P and Q
Mathematically, we can use the internal division formula.
In this case, the coordinates of y is 0 in all cases
So the coordinates of P is (-5,0) while the coordinates of Q is (7,0)
Now, the coordinates of X divides the line PQ in the ratio 5 to 6
Using the internal divison formula, we have;
[tex](x,y)\text{ = }\frac{mx_2+nx_1}{m+\text{ n}},\text{ }\frac{my_2+ny_1}{m+\text{ n}}[/tex]In this case however, we are going to focus on the x-axis part of the question since the values of y at all points is 0
m , n are the division values which are 5 and 6 respectively in this case
x2 is 7 while x1 is -5
Substituting all of these, we have;
[tex]\begin{gathered} (x,y)\text{ = }\frac{5(7)\text{ + 6(-5)}}{11},\text{ 0} \\ \\ (x,y)\text{ = }\frac{35-30}{11},\text{ 0} \\ (x,y)\text{ = }\frac{5}{11},\text{ 0} \end{gathered}[/tex]So the coordinate of point X which is 5/6 of the distance between P and Q is 5/11
Find the common difference and the recursive formula. 22,19,16,13
The common difference between each term is -3.
19 - 22 = -3
16 - 19 = -3
13 - 16 = -3
The recursive formula of an arithmetic sequence follows the pattern below:
[tex]a_n=a_{n-1}+d,n\ge2[/tex]where d = common difference and number of terms "n" must be more than or equal to two.
To be able to get the recursive formula, we will plug in the common difference assuming that first term a₁ = 22. Therefore, the recursive formula is:
[tex]a_n=a_{n-1}-3,for\text{ n}\ge2[/tex]Shaq was climbing a cliff. He stopped for a snack. After that, he slipped 20 feet to an earlier foothold and then slipped 4 feet to another foothold. Model the distance Shaq traveled on the cliff after his snack as a sum.
1) Let's make a sketch to better understand this:
Suppose Shaq was on 40 feet after the snack he slipped 20 feet and then 6 feet.
After the snack, He traveled 20 +4 = 24 feet
From the initial point, He went 24 feet down.
every week, Hector works 20 hours and earns $210.00. he eans a constant amount per hour. write an equation that can be used to determine the number of hours, h, Hector works given the number of weeks, w.
From the question, we're told that Hector earns $210.00 for working 2hours every week. Let's go ahead and determine
3. At which of the following angles is the tangent function undefined?(1) 0 =180°(3) 0 = 45°(4) 0 =-360°(2) 0=-90°
The correct answer is angle 90 degrees.
Explanation:
The tangent of angle 90 degrees is undefined.
[tex]undefined[/tex]. Calculate: (81)3/2
Calculate:
[tex]81^{\frac{3}{2}}[/tex]The fractional exponent can be written as:
[tex]\sqrt[2]{81^3}=(\sqrt[]{81})^3[/tex]The square root of 81 is 9, thus:
[tex](\sqrt[]{81})^3=9^3=729[/tex]What do you notice about the measures of the sides or the measures of angles that form triangles?
The angles sum up to give 180°
Only one of the angles can be an obtuse angle, we can;t have two bothuse angle in a triangle. BUT we can have two acute angles and one obtuse angle in a triangle.
We can also have a 90 degree and 2 acute angle in a triangle.
Examples
The angles sum up to give 180°
Only one of the angles can be an obtuse angle, we can;t have two bothuse angle in a triangle. BUT we can have two acute angles and one obtuse angle in a triangle.
We can also have a 90 degree and 2 acute angle in a triangle.
Examples
I wills send you a picture
Draw the tank
we can use the formula of the volume of a cylinder
[tex]V=\pi\times r^2\times h[/tex]we can repalce the value of the volume (320pi) and the height
[tex]\begin{gathered} 320\pi=\pi\times r^2\times20 \\ 320\pi=20r^2\pi \end{gathered}[/tex]now solve for r^2 dividing 20pi on both sides
[tex]\begin{gathered} \frac{320\pi}{20\pi}=r^2 \\ \\ r^2=16 \\ \end{gathered}[/tex]and solve for r using roots
[tex]\begin{gathered} r=\sqrt[]{16} \\ \\ r=4 \end{gathered}[/tex]the value of the radious is 4ft and the diameter double, then
[tex]\begin{gathered} d=2\times4 \\ d=8 \end{gathered}[/tex]diameter of the cylinder is 8 ft then rigth option is C
A small regional carrier accepted 23 reservations for a particular flight with 2o seats. 14 reservations went to regular customers who will arrive for the flight. each of the remaining passengers will arrive for the flight with a 50 % chance ,independently of each other. (answers accurate to 4 decimal places.) Find the probability that overbooking occurs find the probability that the flight has empty seats
Let's begin by identifying key information given to us:
Number of seats = 20
Number of reservation = 23
14 regular customers show up. So, we have:
[tex]23-14=9RemainingCustomers[/tex]The number of seats left is:
[tex]20-14=6seats[/tex]Overbooking means that more than 6 remaining customers show up (that could mean 7 or 8 or 9 of the remaining customers show up)
The probability of more than 6 customers arriving is given by:
I need help with this practice problem I’m having trouble solving it
A generic cosecant function is
[tex]f(x)=A\csc (kx+\theta)+C[/tex]We must find A, k, θ, and C using the information that we have.
Finding A:
To find A we can use the range of the function, we know there is a gap between -9 and 5, that's the crucial information, the value of A will be the mean of |-9| and |5| (in modulus!), therefore
[tex]A=\frac{|-9|+|5|}{2}=\frac{9+5}{2}=\frac{14}{2}=7[/tex]Therefore
[tex]f(x)=7\csc (kx+\theta)+C[/tex]Finding C:
We can use the fact that we know A and find C, let's suppose that
[tex]\csc (kx+\theta)=1[/tex]For an unknown value of x, it doesn't matter, using the range again we can use the fact that 5 is a local minimum of the function, therefore, when the csc(kx + θ) is equal to 1 we have that the function is equal to 5
[tex]\begin{gathered} 5=7\cdot1+C \\ \\ C=-2 \end{gathered}[/tex]And we find that C = -2. Tip: You can also suppose that it's -1 and use -9 = 7 + C, the result will be the same.
Finding k:
Now we will use the asymptotes, we have two consecutive asymptotes at x = 0 and x = 2π, it means that the sin(kx) is zero at x = 0 and the next zero is at x = 2π, we know that sin(x) is zero every time it's a multiple of π, which gives us
[tex]\begin{gathered} \sin (0)=0\Rightarrow\sin (k\cdot0)=0\text{ (first zero | first asymptote)} \\ \sin (\pi)=0\Rightarrow\sin (2k\pi)=0\Rightarrow k=\frac{1}{2}\text{ (second zero | second asymptote)} \end{gathered}[/tex]Therefore, k = 1/2
[tex]f(x)=7\csc (\frac{x}{2}+\theta)-2[/tex]Finding θ:
It's the easiest one, since we have a zero at x = 0 it implies that θ = 0
Therefore our function is
[tex]f(x)=7\csc (\frac{x}{2})-2[/tex]Final answer:
[tex]f(x)=7\csc \mleft(\frac{x}{2}\mright)-2[/tex]
3(-4+x)<-33 I need to solve for x
Simplify the inequality.
[tex]\begin{gathered} \frac{3(-4+x)}{3}<-\frac{33}{3} \\ -4+x+4<-11+4 \\ x<-7 \end{gathered}[/tex]So answer is x<-7.
For the equation y = -2x + 1 A) complete the Table: X l Y -4 04B) Use the appropriate tool to graph the given equation
ANSWER:
a)
b)
EXPLANATION:
Given:
[tex]y=-2x+1[/tex]a) When x = -4, let's go ahead and solve for y;
[tex]\begin{gathered} y=-2(-4)+1 \\ y=8+1 \\ y=9 \end{gathered}[/tex]When x = 0, let's go ahead and solve for y;
[tex]\begin{gathered} y=-2(0)+1 \\ y=0+1 \\ y=1 \end{gathered}[/tex]When x = 4, let's go ahead and solve for y;
[tex]\begin{gathered} y=-2(4)+1 \\ y=-8+1 \\ y=-7 \end{gathered}[/tex]b) Using the above values, we can go ahead and the equation as seen below;
Sports Authority marks up New Balance sneakers $30 and sells them for $109. Markup is on cost. What are the cost and percent markup?
Answer: $79 and percentage is 36%
Classifying systems of linear equations from graphsFor each system of linear equations shown below, classify the system as "consistent dependent," "consistent independent," or "inconsistent." Then, choose thebest description of its solution. If the system has exactly one solution, give its solution.System ASystem B System C
Consistent dependant system- System B: It has infinite number of solutions, in this case, the graphs of the lines are the same.
Consistent independent system- System C: It has exactly one solution. In this case, both lines cross each other at exactly one point.
Solution : (-2,-2)
Inconsistent: System A.
When a system has no solution, lines never cross each other.
What is this sign of 30ﾟ angle and the sign of the 60ﾟ angle
We are asked to find out the values of sine 60° and sine 30°
Recall from the trigonometric ratios,
[tex]\sin \theta=\frac{\text{opposite}}{\text{hypotenuse}}[/tex]From the given triangle,
With respect to angle 60°, the opposite side is 25√3 ft and the hypotenuse is 50 ft.
Let us substitute these values into the above sine ratio
[tex]\begin{gathered} \sin \theta=\frac{\text{opposite}}{\text{hypotenuse}} \\ \sin 60\degree=\frac{25\sqrt[]{3}}{50} \\ \sin 60\degree=\frac{\sqrt[]{3}}{2} \end{gathered}[/tex]So, the value of sine 60° is √3/2
From the given triangle,
With respect to angle 30°, the opposite side is 25 ft and the hypotenuse is 50 ft.
Let us substitute these values into the above sine ratio
[tex]\begin{gathered} \sin \theta=\frac{\text{opposite}}{\text{hypotenuse}} \\ \sin 30\degree=\frac{25}{50} \\ \sin 30\degree=\frac{1}{2} \end{gathered}[/tex]So, the value of sine 30° is 1/2
Therefore, the sine of 60ﾟ angle is √3/2 and the sine of 30ﾟ angle is 1/2
A is the incenter of Triangle FHG Find the length of AT. Explain your thinking.
we have that
The incenter is the center of the triangle's incircle, the largest circle that will fit
AR=AT=AS -----> radius of the inscribed circle in the triangle
therefore
AT=3 units
the product of 12 and 3 decreased by 6
the product of 12 and 3 decreased by 6
we have that
the product of 12 and 3 ------> is number 12 multiplied by 3
12*3
decreased by 6
12*3-6=36-6=30
the answer is 30determine whether the equation defines y as function of x
To answer this question, we need to solve the equation for y in the third case:
[tex]3x+2y=5\Rightarrow2y=5-3x\Rightarrow y=\frac{5}{2}-\frac{3}{2}x\Rightarrow y=-\frac{3}{2}x+\frac{5}{2}[/tex]We can see from this case that for every value of x, there must be a value in y, and this is the main condition for a relationship to be a function. Then, y is a function of x.
In the fourth case, we have a similar case, for every possible value of x, there must be a value for y. Then, y is a function of x.
As we can see, the red graph is for the linear equation and the black one is for the one with the radical ( y = -sqrt(x+1)).
If we pass a vertical line to either function (alone), we will have only a point that passes through this vertical line, and with this graphical information, we can also say that both are functions of y (for each case).
57. do not use the answer under the line in the explanation itself, only refer to it to make sure of your work. USE DERIVITIVES NOT GRAPHING
Explanation
Question 57
[tex]\:f\left(x\right)=2x^3-15x^2+24x[/tex]To find the extreme values
[tex]\begin{gathered} \mathrm{Suppose\:that\:}x=c\mathrm{\:is\:a\:critical\:point\:of\:}f\left(x\right)\mathrm{\:then,\:} \\ \mathrm{If\:}f\:'\left(x\right)>0\mathrm{\:to\:the\:left\:of\:}x=c\mathrm{\:and\:}f\:'\left(x\right)<0\mathrm{\:to\:the\:right\:of\:}x=c\mathrm{\:then\:}x=c\mathrm{\:is\:a\:local\:maximum.} \\ \mathrm{If\:}f\:'\left(x\right)<0\mathrm{\:to\:the\:left\:of\:}x=c\mathrm{\:and\:}f\:'\left(x\right)>\:0\mathrm{\:to\:the\:right\:of\:}x=c\mathrm{\:then\:}x=c\mathrm{\:is\:a\:local\:minimum.} \\ \mathrm{If\:}f\:'\left(x\right)\mathrm{\:is\:the\:same\:sign\:on\:both\:sides\:of\:}x=c\mathrm{\:then\:}x=c\mathrm{\:is\:neither\:a\:local\:maximum\:nor\:a\:local\:minimum.} \end{gathered}[/tex]So, we will have the steps below
Step 1:
[tex]\begin{gathered} \mathrm{Plug\:the\:extreme\:point}\:x=0\:\mathrm{into}\:2x^3-15x^2+24x\quad \Rightarrow \quad \:y=0 \\ \mathrm{Minimum}\left(0,\:0\right) \end{gathered}[/tex]Step2:
[tex]\begin{gathered} \mathrm{Plug\:the\:extreme\:point}\:x=1\:\mathrm{into}\:2x^3-15x^2+24x\quad \Rightarrow \quad \:y=11 \\ \mathrm{Maximum}\left(1,\:11\right) \end{gathered}[/tex]Step 3:
[tex]\begin{gathered} \mathrm{Plug\:the\:extreme\:point}\:x=4\:\mathrm{into}\:2x^3-15x^2+24x\quad \Rightarrow \quad \:y=-16 \\ \mathrm{Minimum}\left(4,\:-16\right) \end{gathered}[/tex]Step 4:
[tex]\begin{gathered} \mathrm{Plug\:the\:extreme\:point}\:x=5\:\mathrm{into}\:2x^3-15x^2+24x\quad \Rightarrow \quad \:y=-5 \\ \mathrm{Maximum}\left(5,\:-5\right) \\ \end{gathered}[/tex]Thus, we will have
[tex]\mathrm{Minimum}\left(0,\:0\right),\:\mathrm{Maximum}\left(1,\:11\right),\:\mathrm{Minimum}\left(4,\:-16\right),\:\mathrm{Maximum}\left(5,\:-5\right)[/tex]Hence, our answer is
[tex]\begin{gathered} \begin{equation*} \mathrm{Minimum}\left(4,\:-16\right) \end{equation*} \\ \begin{equation*} \mathrm{Maximum}\left(1,\:11\right) \end{equation*} \end{gathered}[/tex]All the formation your name is on the picture picture provided
The range of the data is the difference between the maximum data value and the minimum.
In a box plot, the maximum and the minimum are indicated by the dots at the end of the horizontal line.
Here,
Maximum = 10
Minimum = 4.5
Thus, the range of the data is:
[tex]Range=10-4.5=5.5[/tex]A volleyball drops 8 meters and bounces up 2 meters.
Use the expression |-8 + 2 to find the total distance
the volleyball travels. The total distance the volleyball travels is
✓meters.
The volleyball travelled a total distance of 10 meters
How to determine the total distance travelled by the volleyball?From the question, the given parameters are
Initial height = 8 meters
Height of bounce = 2 meters
The expression of the total distance is represented as
Total distance = |-8| + 2
Remove the absolute symbol
Total distance = 8 + 2
Evaluate the sum
Total distance = 10
Hence, the total distance travelled by the volleyball is 10 meters
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Jane, Chau, and Deshaun have a total of $82 in their wallets. Deshaun has 2 times what Jane has. Chau has $6 less than Jane. How much does each have?
Jane, Chau, and Deshaun have $22, $16 and $44 respectively in their wallets.
let x represent represent the amount of Jane.
let Y represent represent the amount of chau.
let z represent represent the amount of Deshaun.
Jane, Chau, and Deshaun have a total of $82" can be represented as
x + y + z = $82 .......(1)
Deshaun has 4 times what Jane has. It can be represented mathematically as
y = 2x .......(2)
Chau has $6 less than Jane. It can be represented mathematically as
z= x- 6 .......(3)
we can now solve the equations using the substitution method
substitute equation (2) and (3) into equation (1)
x + y + z = $82
x + 2x + x-6 = $82
4x -6 = $82
4x - 6 = $82
4x = $82 + 6
4x = 88
x = $22
from equation 2
y = 2x
y = 2 x 22 = $44
z = x- 6
z = 22 - 6
z=$16
Jane, Chau, and Deshaun have $22, $16 and $44 respectively in their wallets.
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Find the vertex of the following equation: y = -5x² - 270x - 520
In order to find the vertex of this quadratic equation, first let's find the coefficients a, b and c from the standard form of the quadratic equation:
[tex]y=ax^2+bx+c[/tex]Comparing with the given equation, we have a = -5, b = -270 and c = -520.
Now, let's calculate the x-coordinate of the vertex using the formula below:
[tex]\begin{gathered} x_v=\frac{-b}{2a} \\ x_v=\frac{-(-270)}{2\cdot(-5)} \\ x_v=\frac{270}{-10} \\ x_v=-27 \end{gathered}[/tex]Using this value of x in the equation, we can find the y-coordinate of the vertex:
[tex]\begin{gathered} y_v=-5x^2_v-270x_v-520 \\ y_v=-5\cdot(-27)^2-270\cdot(-27)-520 \\ y_v=-5\cdot729+7290-520 \\ y_v=-3645+7290-520 \\ y_v=3125 \end{gathered}[/tex]Therefore the vertex is located at (-27, 3125).