To prove that the order of u(n) is even when n > 2, we can use Corollary 2 of Lagrange's theorem (Theorem 7.1). Corollary 2 states that if G is a group and a is an element of G of finite order, then the order of a divides the order of G.
Let's consider the group G = U(n), the multiplicative group of integers modulo n, and let a = u(n), an element of G. We want to show that the order of a is even when n > 2.
By definition, the order of an element a in a group is the smallest positive integer k such that a^k = e, where e is the identity element of the group.
Since a = u(n), we have a^n ≡ 1 (mod n) by Euler's theorem. This implies that a^n - 1 is divisible by n.
Now, let's consider the order of a. Assume the order of a is odd, i.e., k is an odd positive integer such that a^k = e. This implies that a^(2k) = (a^k)^2 = e^2 = e.
Since k is odd, 2k is even. Therefore, we have found a positive integer (2k) such that a^(2k) = e, contradicting the assumption that k is the smallest positive integer satisfying a^k = e. Thus, the order of a cannot be odd.
By Corollary 2 of Lagrange's theorem, the order of a divides the order of G. Since n > 2, the order of G is even (it contains the identity element and at least one non-identity element). Therefore, the order of a (u(n)) must also be even.
Hence, we have proven that the order of u(n) is even when n > 2 using Corollary 2 of Lagrange's theorem.
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The data below show the hourly wages earned by six employees. The second column shows the number of years of work experience for each employee. The third column is a binary variable indicating whether or not the employee has completed a training program. For the following questions, you may check your work using a calculator but must show how you set up the expression you are using (i.e. show your work).
wage experience training
8 1 0
20 10 0
10 2 0
12 4 0
16 4 1
14 6 1
a) What are the mean, variance, and standard deviation of wages?
b) What is the mean, variance, and standard deviation of experience?
c) Draw a scatter plot with experience on the horizontal axis and wages on the vertical axis. Draw a straight line that best fits the data points. Based on this line, what is the approximate wage of someone with 0 years of experience?
d) What is the covariance of wages and experience? Is this consistent with the line you drew in part c)? Explain.
e) What is the correlation of wages and experience?
The approximate wage of someone with 0 years of experience is -4.64
a) What are the mean, variance, and standard deviation of wages?The dataset of wages is
Wages = 8, 20, 10, 12, 16, 14
The mean is calculated as:
Mean = Sum/Count
This gives
Mean = (8+ 20+ 10+ 12+ 16+ 14)/6
Mean = 13.33
The variance is
Variance = ∑(x- mean)²/n-1
This gives
Variance = [(8 - 13.33)^2 + (20 - 13.33)^2 + (10 - 13.33)^2 + (12 - 13.33)^2 + (16 - 13.33)^2 + (14 - 13.33)^2]/5
Evaluate
Variance = 18.67
The standard deviation is
Standard deviation = √Variance
This gives
Standard deviation = √18.67
Evaluate
Standard deviation = 4.32
b) What is the mean, variance, and standard deviation of experience?The dataset of experience is
Wages = 1, 10, 2, 4, 4, 6
The mean is calculated as:
Mean = Sum/Count
This gives
Mean = (1+ 10+ 2+ 4+ 4+ 6)/6
Mean = 4.5
The variance is
Variance = ∑(x- mean)²/n-1
This gives
Variance = [(1 - 4.5)^2 + (10 - 4.5)^2 + (2 - 4.5)^2 + (4 - 4.5)^2 + (4 - 4.5)^2 + (6 - 4.5)^2]/5
Evaluate
Variance = 10.3
The standard deviation is
Standard deviation = √Variance
This gives
Standard deviation = √10.3
Evaluate
Standard deviation = 3.21
c) Draw a scatter plot with experience on the horizontal axis and wages on the vertical axis.See attachment for the scatter plot and the straight line that best fits the data points.
The equation of the straight line is
y = 0.69x - 4.64
The approximate wage of someone with 0 years of experience is
y = 0.69 * 0 - 4.64
Evaluate
y = -4.64
Hence, the approximate wage of someone with 0 years of experience is -4.64
d) What is the covariance of wages and experience?This is calculated as:
Cov(x,y) = ∑(x - x mean)(y - y mean)/N - 1
So, we have:
Cov(x,y) = [(8 - 13.33) * (1 - 4.5) + (20 - 13.33)*(10 - 4.5) + (10 - 13.33)*(2 - 4.5) + (12 - 13.33)*(4 - 4.5) + (16 - 13.33)*(4 - 4.5) + (14 - 13.33)*(6 - 4.5)]/5
Evaluate
Cov(x,y) = 12.80
Hence, the covariance of wages and experience is 12.80
Is this consistent with the line you drew in part c?
Yes it is
e) What is the correlation of wages and experience?Here, we use a graphing calculator.
From the graphing calculator, we have:
r = 0.9231
Hence, the correlation of wages and experience is 0.9231
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The pressure, p, with which water passes through a pipe varies inversely as the square of the pipe's radius, 2. If k is the constant of
variation, which equation represents this situation?
Opr²-k
Op=r²2²+k
Op+r²2²=k
Op=kr²
The variation equation is p = kr^2
How to determine the variation equation?The variables are give as:
Pressure = P
Radius= r
The direct variation from p to the square of r is represented as:
[tex]p\ \alpha\ r^2[/tex]
Express as an equation
p = kr^2
Where k represents the constant of variation
Hence, the variation equation is p = kr^2
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Answer:
The correct answer is the 3rd option
The height of a certain species of plants is uniformly distributed on the interval of 5-12 cm. a plant was measured by jack. what is the probability that the height of this plant is between 6.3 and 7.8cm?
21 % is the answer.
Problem is that the height of this plant 6.3 and 7.8 cm
P (6.3 <x< 7.8)
12 -5 = 7cm
7.8 - 6.3 = 1.5cm
1.5 / 7 = 0.21
=21%
Probability is a field of mathematics that deals with the numerical description of the likelihood that an event will occur or that a statement is true. The probability of an event is a number between 0 and 1, with approximately 0 indicating the impossibility of the event and 1 indicating certainty.
Probability is a measure of the likelihood that an event will occur. Many events cannot be predicted with absolute certainty. You can only predict the probability that an event will occur.
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Someone please answer this for me (Algebra 2)
Using division of fractions, the equivalent expression is:
B. [tex]\frac{4}{d - 6}[/tex].
How to divide fractions?To divide two fractions, we multiply the first by the inverse of the second.
In this problem, we have that:
The first fraction is: [tex]\frac{2d - 6}{d^2 + 2d - 48} = \frac{2(d - 3)}{(d + 8)(d - 6)}[/tex].The second fraction is: [tex]\frac{d - 3}{2d + 16} = \frac{d - 3}{2(d + 8)}[/tex].Applying the multiplication, we have that:
[tex]\frac{2(d - 3)}{(d + 8)(d - 6)} \times \frac{2(d + 8)}{d - 3} = \frac{4}{d - 6}[/tex].
Hence option B is correct.
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Using the unit circle, determine the value of sin(-240°).
Answer:
See my picture below for the answer.
Step-by-step explanation:
If you look at the unit circle and locate 240 degrees. You will see an ordered pair. The first number -1/2 is the cos of 240 and the second number is the sin.
Answer:
it’s the square root of 3, divided by 2 (sorry, can’t type any special characters at the moment)
Step-by-step explanation:
-240 = 120 because you measure negative angles clockwise starting at 0.
Using unit circle, value of sin for an angle is the 2nd number
It takes the pit crew 15 seconds to change a
tire. How long is this in minutes?
Answer:
Hey there :) The answer is 0.25 minutes.
Step-by-step explanation:
1 minute = 60 seconds
So, 15 ÷ 60 = 0.25 minutes
Select the correct answer. malik is baking pumpkin bread and banana bread for friends and family. his pumpkin bread recipe calls for 4 eggs and cups of flour, and his banana bread recipe calls for 1 egg and cups of flour. malik has 14 eggs, 16 cups of flour, and plenty of other ingredients to make multiple loaves. what is one combination of breads malik can bake without getting more ingredients? a. 3 loaves of pumpkin bread and 3 loaves of banana bread b. 1 loaf of pumpkin bread and 9 loaves of banana bread c. 2 loaves of pumpkin bread and 6 loaves of banana bread d. 5 loaves of pumpkin bread and 1 loaf of banana bread
The correct answer is
2 loaves of pumpkin bread and 6 loaves of banana bread. Option C.
How to find a possible combination of bread?For the solution for the Pumpkin bread:
4 eggs
3 1/2 cups of flour
Banana bread:
1 egg
1 (0.5) cups of flour
Available ingredients:
14 eggs
16 cups of flour
Assume 2 loaves of pumpkin bread and 6 loaves of banana bread
For the solution for the Pumpkin bread:
PB=4 eggs* 2
PB= 8 eggs
PB=3 *0.5 * 2
PB = 7 cups of flour
For the solution of the Banana bread:
BB =1 *6
BB= 6 eggs
For cups
BB= 1 (0.5) * 6
BB= 9 cups of flour
In conclusion, The solution of the Total used ingredients
I=8 + 6
I= 14 eggs
For cups
IC=7 + 9
IC= 16 cups of flour
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Answer:
C 2 loaves of Pb and 6 loaves of BB
Step-by-step explanation:
Which ordered pair comes from the table?
X Y
3 4
4 4
5 2
6 5
A. (4,3)
B. (2,5)
C. (5,6)
D. (6,5)
The ordered pairs from the options given that is from the table is: D. (6,5).
How to Write an Ordered Pair from a Table?An ordered pair is written in the form, (x, y). The first number in the bracket is the x-value on the table, and the second number is its corresponding y-value form the table.
From the table given, the ordered pairs are: (3, 4), (4, 4), (5, 2), and (6, 5).
The number pair from the table from the given options is: D. (6,5).
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3.) Solve 5y +1 <36
Answer:
y<7
Step-by-step explanation:
Please mark brainliest and have a great day!
Answer: y < 7
Step-by-step explanation:
We can assume that the less than sign is an equals sign and use algebra to isolate y.
[tex]5y+1-1 < 36-1\\ 5y < 35\\\frac{5y}{5} < \frac{35}{5}\\ y < 7[/tex]
We first subtracted 1 from both sides to get rid of the +1.
Then we divided both sides by 5 to isolate y.
Which is the correct equation of the line for the following graph?
Answer:
(c) y = -3x +6
Step-by-step explanation:
The equation can be written in slope-intercept form when we know the slope and the y-intercept.
Slope-intercept formThe slope-intercept form of the equation of a line is ...
y = mx +b . . . . . . where m is the slope and b is the y-intercept
Y-interceptThe y-intercept is the y-value where the line crosses the y-axis. It is 6. This eliminates choices A and D, leaving choices ...
y = 3x +6 . . . . . choice By = -3x +6 . . . . .choice CSlopeThe line goes down to the right. This is a negative slope, eliminating choice B.
The equation of the line is y = -3x +6.
__
Additional comment
The line intersects the x-axis at x=2. This means the "rise" of the line is -6 units for a "run" of 2 units. The numerical value of the slope is ...
m = rise/run = -6/2 = -3
Now, we know the parameters of the equation are m=-3, b=6, and the equation of the line is
y =-3x +6
WILL GIVE BRAINLIEST!!!!!!!!!
Considering the given proportions, the correct statements are:
A person in the group who is more likely to read is a boy.A person in the group who is more likely to listen to music is a girl.What is a proportion?A proportion is a fraction of a total amount, and the measures are related using a rule of three.
The table gives these following proportions:
Boys and girls regarding books, boys are likelier to prefer books.Boys and girls regarding music, girls are likelier to prefer music.Hence the correct statements are:
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The formula for the area of a triangle is Area=2(base)(height). Write an
expression for the height of a triangle that has an area equal to 10x²+20x and a
base equal to 20x.
Answer:
B
Step-by-step explanation:
area = 10x^2 + 20x
base = 20x
10x^2 + 20x = 0.5(20x)(height)
10x^2 + 20x = 10x(height)
(10x^2 + 20x) / 10x = height
[tex]\frac{10x^2 + 20x}{10x} = \frac{10x + 20}{10} = \frac{x + 2}{1} = x + 2[/tex]
The answer is B
A couple quick algebra 1 questions for 50 points!
Only answer if you know the answer, quick shout-out to Dinofish32, tysm for the help!
Answer:
8. yes; 18
9. no
10. no
11. yes; -9
Step-by-step explanation:
The equation for inverse variation is y = k/x. The constant k will be the product of x and y: k = xy.
8.Products are (-3)(-6) = (2)(9) = (4.5)(4) = 18.
Inverse variation with k = 18.
9.Products are ...
(5)(4) ≠ (10)(8).
Not inverse variation.
10.Products are ...
(2)(25) ≠ (2.5)(16).
Not inverse variation.
11.Products are (3)(-3) = (9)(-1) = (12)(-.75) = -9.
Inverse variation with k = -9.
find the equation of the straight line with the following gradient point c) 5, (-1,-3)
Answer:
y = 5x + 2
Step-by-step explanation:
Standard form of Equation of line: y = mx + c
where m = slope or gradient,
c = y-intercept
From the question we know that:
x = - 1
y = - 3
m = 5
We will substitute these values into the equation to find c.
- 3 = (5)(- 1) + c
- 3 = - 5 + c
c = - 3 + 5 = 2
Therefore the equation of this line is: y = 5x + 2
division of 3 fraction
(2/5 ÷ 3/8 ) ÷ -3/5
Answer:
-16/9
Step-by-step explanation:
( [tex]\frac{2}{5}[/tex] ÷ [tex]\frac{3}{8}[/tex] ) ÷ [tex]-\frac{3}{5}[/tex]
= ( [tex]\frac{2}{5} * \frac{8}{3}[/tex] ) ÷ [tex]-\frac{3}{5}[/tex]
= [tex]\frac{16}{15}[/tex] ÷ [tex]-\frac{3}{5}[/tex]
= [tex]\frac{16}{15} * -\frac{5}{3}[/tex]
= -16/9
QUICK HELP PLSSSS
which shape is the cross section of a square pyramid
Answer:
B
Step-by-step explanation:
when a square pyramid is cross sectioned the shape that will appear is the same as that of the base
Solve for X
7(8x + 6) = -1
9(5+ x) = 15 (10+ x)
Answer: 7(8x + 6) = -1: Exact form: x = -43/56, decimal form: x = -0.76785714
Thats all i have because you can't solve for x for the second one!
Determine the equation of a line that passes through (-2,5) and is parallel to the line whose equation is 5y +2x = 10.
Answer:
Step-by-step explanation:
Givens
line: 5y + 2x = 10
point (-2 , 5)
Discussion and Solution
You have to rearrange the given line to get the slope. It isn't nice, but it's not impossible.
5y + 2x = 10 Subtract 2x from both sides
5y + 2x - 2x = -2x + 10 Simplify
5y = - 2x + 10 Divide by 5
5y/5 = -2x/5 + 10/5 Simplify
y = -0.4x + 2
So the new equation is going to have a slope of - 0.4
So far what you have is
y = - 0.4x + b
Now use the point to find the y intercept. What you should be thinking about the point is that when x = -2 then y = 5. Substitute that into the new equation.
5 = -0.4*-2 + b
5 = 0.8 + b Subtract 0.8 from both sides
5 - 0.8 = 0.8 - 0.8 + b Simplify
4.2 = b
Answer
y = -0.4x + 4.2
(2x+5/60-95x-3/30=3/2
Answer:
x = -0.01685
(Is the equation written correctly? This feels like a very unfortunate number)
Step-by-step explanation:
(2x+5/60-95x-3/30=3/2
2x+(5/60)-95x-(3/30)=3/2
2*(2x+(5/60)-95x-(3/30))=3 [Multiply both sides by 2]
4x+(10/60)-190x-(6/30))=3
(2/30)-186x-(6/30))=3 [Simplify]
-186x-(4/30)=3
-186x=3+(4/30)
-186x=(90/30)+(4/30)
x = (94/30)/(-186)
x = -0.01685
Zachary is a salesperson who sells computers at an electronics store. He makes a base pay amount each day and then is paid a commission as a percentage of the total dollar amount the company makes from his sales that day. Let P represent Zachary's total pay on a day on which he sells x dollars worth of computers. The table below has select values showing the linear relationship between x and P. Determine the percentage commission Zachary makes on his daily sales.
The tabular values can be used to derive the formula relationship between x and P which gives the percentage commission of Zachary as 1%
How can the percentage commission be calculated?From the above table, we have;
P = x × y% + bWhere;
y = The fixed percentage commission Zachary makes
b = Zachary's base pay
Therefore;
105 = 3000 × y + b... (1)110 = 3500 × y + b... (2)125 = 5000 × y + b... (3)Subtracting equation (1) from (2), gives;
110 - 105 = (3500 - 3000) × y + b - b = 0
5 = 500•y
y = 5/500 = 0.01 = 1%
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Which of the following equations describes the graph?
Identify a transformation of the function f(x) = x by observing the equation of the function g(x) = x − 90.
The base graph has been shifted down 90 units.
The base graph has been shifted right 90 units.
The base graph has been shifted left 90 units.
The base graph has been shifted up 90 units.
Using translation concepts, the change of f(x) = x to g(x) = x - 90 can be described as follows:
The base graph has been shifted down 90 units.
What is a translation?A translation is represented by a change in the function graph, according to operations such as multiplication or sum/subtraction in it's definition.
In this problem, we have that g(x) = f(x) - 90, that is, the base graph was shifted down 90 units.
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Solve for w, where w is a real number.
Answer:
w= 9
Step-by-step explanation:
[tex] \sqrt{ - 4w + 61} = w - 4[/tex]
Square both sides:
-4w +61= (w -4)²
[tex]\boxed{(a - b)^{2} = a^2 -2ab + b^2 }[/tex]
Expand:
-4w +61= w² -2(w)(4) +4²
-4w +61= w² -8w +16
Simplify:
w² -8w +16 +4w -61= 0
w² -4w -45= 0
Factorize:
(w -9)(w +5)= 0
w -9= 0 or w +5= 0
w= 9 or w= -5 (reject)
Note:
-5 is rejected since we are only taking the positive value of the square root here. If the negative square root is taken into consideration, then w= -5 would give us -9 on both sides of the equation.
Why do we use negative square root?
When solving an equation such as x²= 4,
we have to consider than squaring any number removes the negative sign i.e., the result of a squared number is always positive.
In the case of x²= 4, x can be 2 or -2. Thus, whenever we introduce a square root, a '±' must be used.
However, back to our question, we did not introduce the square root so we should not consider the negative square root value.
Answer: w=9.
Step-by-step explanation:
[tex]\sqrt{-4w+61}=w-4[/tex]
Tolerance range:
[tex]\left \{ {{-4w+61\geq 0} \atop {w-4\geq 0}} \right. \ \ \ \ \left \{ {{4w\leq 61\ |:4} \atop {w\geq 4}} \right. \ \ \ \ \left \{ {{w\leq 15,25} \atop {w\geq 4}} \right. \ \ \ \ \Rightarrow\ \ \ \ \\w\in[4;15,25].[/tex]
Solution:
[tex](\sqrt{-4w+61})^2=(w-4)^2\\-4w+61=w^2-2*w*4+4^2\\-4w+61=w^2-8w+16\\w^2-4w-45=0\\D=(-4)^2-4*1*(-45)\\D=16+4*45\\D=16+180\\D=196.\\\sqrt{D}=\sqrt{196}\\\sqrt{D}=14 \\w_{1,2}=\frac{-(-4)б14}{2} \\w_1=\frac{4-14}{2} \\w_1=\frac{-10}{2} \\w_1=-5\ \notin tolerance\ range.\\w_2=\frac{4+14}{2}\\ w_2=\frac{18}{2} \\w_2=9\ \in tolerance \ range.[/tex]
The two-way frequency table below shows data on playing a sport and playing a musical instrument for students in a class.
Complete the following two-way table of row relative frequencies.
(If necessary, round your answers to the nearest hundredth.)
Here is the completed table:
Plays a sport Doesn't play a sport
Plays a musical instrument 0.46 0.54
Doesn't play a musical instrument 0.73 0.27
What are the row frequencies?
Relative frequency measures how often a value appears relative to the sum of the total values.
Plays a sport Doesn't play a sport
Plays a musical instrument (6/13) = 0.46 (7/13) = 0.54
Doesn't play a musical instrument (8/11) 0.73 (3/11) =0.27
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help me i kinda stuck
For the given data,
Mean = 2.12
Median = 2
Mode = 2
Range = 3
Statistics
From the question we are to determine the mean
From the given table
Scoops of Ice cream Male Female Other Total frequencies
1 7 3 3 13
2 8 11 2 21
3 10 1 0 11
4 2 1 1 4
∴ [tex]Mean = \frac{(1 \times 13) + (2 \times 21) +(3 \times 11) + (4 \times 4)}{13 +21+11+4}[/tex]
[tex]Mean = \frac{(13) + (42) +(33) + (16)}{49}[/tex]
[tex]Mean = \frac{104}{49}[/tex]
Mean = 2.12
Median = 2
Mode = 2
Range = 4 - 1 = 3
Hence, for the given data
Mean = 2.12
Median = 2
Mode = 2
Range = 3
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Which of the following terms are possible in the domain of a sequence? Select all that apply.
The terms that are possible on the domain of a sequence are given as follows:
13, 25, 60.
What is a sequence?A sequence is a list containing numbers, that may be related or not. It has a 1st term, 2nd term, and so on until the nth term.
The domain is the indexes of the terms, which starts at one and goes until n, hence the possible numbers are:
13, 25, 60.
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Can someone help me with these problems and show work please !
The domain of g(x)= 3x^2 is Real number and it's range is {y∈R / y≥0}
The domain of h(x)= -3x^2 is Real number and it's range is {y∈R / y ≤0}
The domain of a function f(x) is the set of all values for which the function is defined, and the domain of a function is the set of all values that f takes on. The domain and range are defined for a relation and are the sets of all x-coordinates and all y-coordinates of ordered pairs.
The graph of f(x)= x^2 is given
We need to find the domain and range of g(x)=3x^2 and h(x)= -3x^2
The Graph of 3x^2 is different from x^2 in terms of plotting
When ,
x= -2 g(x)=12
x=-1 g(x)=3
x=0 g(x)=0
x=1 g(x)=3
x=2 g(x)=12
The Graph of -3x^2 is different from x^2 in terms of plotting
When ,
x= -2 g(x)=-12
x=-1 g(x)= -3
x=0 g(x)=0
x=1 g(x)= -3
x=2 g(x)=-12
Domain of g(x) is R and it's range is {y∈R / y≥0}
Domain of h(x) is R and it's range is {y∈R / y ≤0}
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What is the quotient?
StartFraction t + 3 Over t + 4 EndFraction divided by (t squared + 7 t + 12)
(t + 3) squared
(t + 4) squared
StartFraction 1 Over (t + 4) squared EndFraction
StartFraction 1 Over (t + 3) squared EndFraction
The answer choice which represents the quotient of the expression given is; 1/(t+4)².
Which answer choice represents the quotient?It follows from the task content that the quotient which is to be evaluated is;
(t+3)/t+4) ÷ (t²+7t +12)
The quotient can therefore be evaluated as follows;
= (t+3)/t+4) ÷ (t+3)(t+4)
= (t+3)/t+4) × 1/(t+3)(t+4)
= 1/(t+4) × 1/(t+4)
= 1/(t+4)².
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Answer:
c
Step-by-step explanation:
c
20POINTS
help me please
Answer:
a. 50 students are randomly selected from the school; 11 drive to school
b. 407
Step-by-step explanation:
a. to get an accurate viewing, use random people that share 1 large common denominator (from the same school) that also has to do with the question
b. 11/50 is only between 50 students. make the denominator the same (1850) and multiply the same to the numerator
1850/50 = 37
11/50 x 37 = 407/1850
PLEASE HELP ME AS SOON AS POSSIBLE
Answer: [tex]f^{-1}(\text{x}) = (\text{x}-9)^2+4[/tex] when [tex]\text{x} \ge 9[/tex]
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Work Shown:
[tex]f(\text{x}) = 9 + \sqrt{\text{x} - 4}\\\\\text{y} = 9 + \sqrt{\text{x} - 4}\\\\\text{x} = 9 + \sqrt{\text{y} - 4}\\\\\text{x}-9 = \sqrt{\text{y} - 4}\\\\(\text{x}-9)^2 = (\sqrt{\text{y} - 4})^2\\\\(\text{x}-9)^2 = \text{y}-4\\\\(\text{x}-9)^2+4 = \text{y}\\\\f^{-1}(x) = (\text{x} - 9)^2+4[/tex]
Explanation:
I replaced f(x) with y. After that I swapped x and y, then solved for y to get the inverse.
The smallest that [tex]\sqrt{\text{x}-4}[/tex] can get is 0, which means the smallest f(x) can get is 9+0 = 9. The range for f(x) is [tex]\text{y} \ge 9[/tex]
Since x and y swap to determine the inverse, the domain and range swap roles. Therefore, the domain of the inverse [tex]f^{-1}(\text{x})[/tex] is [tex]\text{x} \ge 9[/tex]
So we will only consider the right half portion of the parabola.
The graph is below. The red curve mirrors over the black dashed line to get the blue curve, and vice versa.
Answer:
[tex]f^-^1(x)=(x-9)^2+4[/tex] OR [tex]x^2-18x+85[/tex] if you simplify it
Step-by-step explanation:
to find the inverse function you have to switch x and y with each other and solve for y.
[tex]y=9+\sqrt{x-4}\\x=9+\sqrt{y-4}[/tex] step 1: switch x and y with each other
[tex]x-9=\sqrt{y-4}[/tex] step 2: subtract 9 from both sides
[tex](x-9)^2=\sqrt{y-4}^2[/tex] step 3: square both sides to get rid of the square root
[tex](x-9)^2=y-4\\(x-9)^2+4=y\\[/tex] step 4: add 4 to both sides
you could leave the answer like this or you can simplify to get [tex]x^2-18x+85[/tex]
Someone please helpppp
Answer:
3. Completing the Square
4. 0, 1, or 2
Step-by-step explanation:
Hello!
3) Which of the following is NOT a method to find the x-intercepts of a quadratic equation?a) Graphing
Graphing is a useful method as it lays the answer out in front of you. Once you graph the parabola, you simply take the intersections of the graph and x-axis as your x-intercepts. Therefore, it is a method.
b) Factoring
Factoring can be used to find x-intercepts, and after factoring, you can set each factor to 0 and solve for x, which are the x-intercepts. This method is also called the Zero Product Property.
c) Quadratic Formula
The quadratic formula is meant to solve for the x-intercepts, as the formula uses values of the coefficients to isolate x and solve for it.
Standard form of a Quadratic: [tex]ax^2 + bx + c = 0[/tex]
Quadratic Formula: [tex]x = \frac{-b\pm\sqrt{b^2 - 4ac}}{2a}[/tex]
Plugging in the values of a, b, and c give you the solutions for x.
d) Completing the Square
Completing the square CAN be used to find the x-intercepts, but not directly so. It is used to convert standard form into a different form, known as the vertex form. It is meant so find the vertex, not the x-intercepts.
Therefore, Completing the Square is NOT a method to find the x-intercepts.
4) How many solutions (x-intercepts) can a quadratic equation have?The number and types of solutions depend on the discriminant of the quadratic (the part under the square root in the QF).
Types of roots:
If positive: 2 real rootsIf negative: 2 imaginary roots (technically 0 roots)If equal to 0: 1 real root (double root)That means that a quadratic equation can have 0, 1, or 2 roots.
Answer:
Completing the square and 2, 1, and 0.
Step-by-step explanation:
Person above is correct!!