Find the inverse of the function. f(x) = 7x - 1
The inverse of the function f(x) = 7x - 1 is [tex]f^{-1}(x) = \frac{x}{7} + \frac{1}{7}[/tex]
The given function is:
f(x) = 7x - 1
Make x the subject of the formula
[tex]f(x) = 7x - 1\\\\7x = f(x) + 1\\\\[/tex]
Divide through by 7
[tex]\frac{7x}{7}= \frac{f(x)}{7} + \frac{1}{7}\\\\x = \frac{f(x)}{7} + \frac{1}{7}[/tex]
Replace x by [tex]f^{-1}(x)[/tex] and replace f(x) by x:
[tex]f^{-1}(x) = \frac{x}{7} + \frac{1}{7}[/tex]
Therefore the inverse of the function f(x) = 7x - 1 is [tex]f^{-1}(x) = \frac{x}{7} + \frac{1}{7}[/tex]
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what is the factored form of 64g^3+8?
Answer: For those in the future!
The answer to this is D: (4g+2)(16g^2minus8g+4)
Step-by-step explanation:
Use either the ratio test or the root test as appropriate to determine
The series ∑∞ n=1 an with terms [tex]a_n = (1 * 8 * 15 * ... * (7n-6)) / (7^n * n!)[/tex]diverges. Hence the correct option is 2.
Ratio Test:
For the series ∑∞ n=1 an with terms an, let:
lim n->∞ |an+1 / an| = L
If L < 1, the series converges.
If L > 1, the series diverges.
If L = 1 or the limit diverges, the test is inconclusive.
Applying the Ratio Test:
Here, [tex]a_n = (1 * 8 * 15 * ... * (7n-6)) / (7^n * n!)[/tex]
Let's analyze the limit as n approaches infinity for the ratio of successive terms:
Simplifying the numerator and denominator, we get:
[tex]lim n- > ∞ | 7^(^n^+^1^) * (n+1)! / (7^n * n!) | = lim n- > ∞ | 7 * (n+1) |[/tex]
As n approaches infinity, the constant term 7 becomes insignificant. Therefore:
lim n->∞ |an+1 / an| = lim n->∞ |n+1|
Since the limit approaches positive infinity as n approaches infinity, L > 1.
Conclusion:
Based on the ratio test, the series ∑∞ n=1 an with terms [tex]a_n = (1 * 8 * 15 * ... * (7n-6)) / (7^n * n!)[/tex]diverges. Hence the correct option is 2.
Complete question:
Use either the ratio test or the root test as appropriate to determine whether the series ∑∞, n=1 an with given terms an converges, or state if the test is inconclusive. an= 1⋅8⋅15⋯(7n−6)/7^n n!
The series converges.
The series diverges.
The test is inconclusive.
√7, 2, √8/2
order the number from least to greatest
To order the numbers √7, 2, √8/2 from least to greatest, simplify √8/2 to √2. Then compare √2, 2, and √7 to find the order.
Explanation:To order the numbers √7, 2, √8/2 from least to greatest, first simplify √8/2. We know that √8 = √(4*2) = 2√2, so √8/2 = (2√2)/2 = √2. Now we have √7, 2, and √2. Comparing these numbers, √2 < 2 < √7.
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Use the Social Security tax rate of 6.2% and Medicare tax rate of 1.45% to determine the total deduction.
Gross Pay: $6223
In a coin flip experiment, david decides to flip a coin 50 times. he wants to record the number of times "tails" comes up. using the language of probability, in this experiment, each coin flip is a ________, the way the coin lands is the ________. when the coin comes up "tails", this is considered a ________.
The graph below indicates the distance that the cat (red line) and mouse (green line) have run as a function of the time since the chase began. How far did the mouse run before the cat caught it ?
a. 1 foot c. 22 feet
b. 11 feet d. 6 feet
The mouse ran a distance of 11 feets before being caught by the cat
How to use the graph for our solutionUsing the information on the graph ;
Y - axis = Distance covered by the mouse and cat
X - axis = Time taken to cover Distance
The intersection point of the two lines is the point at which the mouse was caught .
Tracing the point to the y-axis , we get the required Distance value
Hence, from the graph, the distance is 11 feets
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Before being caught by the cat, the mouse sprinted 11 feet.
How can we apply the graph to our solution?
Using the data from the graph;
The distance traveled by the mouse and cat is represented by the Y axis.
X axis = Distance traveled in time
The spot where the two lines connect is where the mouse was trapped.
We get the requisite Distance value by tracing the point to the y-axis.
As a result of the graph, the distance is 11 feet.
estimate the difference between 206 and 167
Find the volume v of the described solid s. a pyramid with height 2h and rectangular base with dimensions 4b and 8b
The volume of a pyramid with a rectangular base and height of 2h is given by the formula V = (1/3) * Base area * Height. The base area for this pyramid is 32b² (derived from 4b * 8b), and the height is 2h. Hence, the volume equates to 64/3 b²h.
Explanation:The volume of a pyramid is calculated with the formula V = (1/3) * Base area * Height. For the pyramid in question, the base is a rectangle with a length of 4b and a width of 8b, so the area of the base (A) is 4b * 8b = 32b². The height of the pyramid is given as 2h. So, by plugging these values into the formula, we obtain the volume of the pyramid as V = (1/3) * 32b² * 2h = 64/3 b²h.
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How can you check the answer to a decimal multiplication problem
Find the product. In the form of a paragraph, explain in complete sentences the steps necessary to simplify the expression and include the final answer in your explanation. Complete your work in the space provided or upload a file that can display math symbols if your work requires it.
( x y 2 z)3 (-9x 2 y z)2
The product of (x y^2 z)^3 and (-9x^2 y z)^2 is found by applying the exponent to each variable inside the parentheses and then multiplying the results together, using exponent rules to simplify. The final simplified expression is 81x^7 y^8 z^5.
Explanation:To find the product of the expressions (x y^2 z)^3 and (-9x^2 y z)^2, we use the properties of exponents. First, we apply the power to each factor inside the parentheses, which means raising each base to the power outside, and then we multiply the corresponding exponents when the same base is found in both terms.
For the first expression (x y^2 z)^3, we get x^3 y^6 z^3, and for the second expression (-9x^2 y z)^2, we have 81x^4 y^2 z^2 after squaring -9 to get 81 and applying the exponent 2 to each variable.
Next, we multiply these results together: x^3 y^6 z^3 × 81x^4 y^2 z^2 = 81x^{3+4} y^{6+2} z^{3+2}, which simplifies to 81x^7 y^8 z^5. The rule of exponent addition when multiplying like bases, x^p x^q = x^{(p+q)}, is applied here to combine the variables with the same base. Our final simplified product is 81x^7 y^8 z^5.
The final answer is [tex]81 x^7 y^8 z^5.[/tex]
To find the product of the expression [tex](x y^2 z)^3 * (-9x^2 y z)^2[/tex], we can follow these steps:
1. Expand the first expression: [tex](x y^2 z)^3 = x^3 (y^2)^3 z^3 = x^3 y^6 z^3.\\[/tex]
2. Expand the second expression: [tex](-9x^2 y z)^2 = (-9)^2 (x^2)^2 y^2 z^2 = 81 x^4 y^2 z^2.[/tex]
3. Multiply the expanded expressions: [tex](x^3 y^6 z^3) * (81 x^4 y^2 z^2) = 81 x^3 x^4 y^6 y^2 z^3 z^2 = 81 x^7 y^8 z^5.[/tex]
Therefore, the final answer is [tex]81 x^7 y^8 z^5[/tex].
The first advertisement discussed in class states that the salespeople make an average of $1,000 per week. Suppose there are nine salespeople. What would the ninth person need to earn for the mean to be $1,000 if the other eight salespeople earned $550, $600, $600, $800, $950, $950, $1,000, and $1,100?
2.6x10^-6 write the folliwing number in standard form (decimal)
Final answer:
To express 2.6x10^-6 in standard decimal form, the decimal point is moved 6 places to the left, giving us 0.0000026.
Explanation:
To write 2.6x10^-6 in standard decimal form, start by understanding the exponent. The notation 10^-6 means that we are moving the decimal point 6 places to the left because it is a negative exponent. Therefore, we start with 2.6 and move the decimal point to the left 6 places, which means we have to add zeros to fill in the gaps.
The final expression in standard form will be 0.0000026.
Explain the solution to |4w + 9| = –7.
i do not get this
Rachel and her friends are making necklaces out of beads and straws. They plan to use 12 beads and 5 straws for each necklace. They have a jar of beads that contains a total of 2,700 beads. How many necklaces can they make?
Write an equation to solve the problem.
you want to buy a CD that cost $18.50. The sales-tax rate in your states is 4.5 percent. How much sales tax will you pay for this CD?
4.5% = 0.045
multiply 18.50 by 0.045 foe the amount of sales tax
18.50 * 0.045 = 0.83
they pay 0.83, which is 83 cents in sales tax
Find the minimum value of the function
Amy has 1/4 hour to do 5 chores. She plans to spend the same fraction of an hour on each chore. She wants to use the number line to help her determine what fraction of an hour she can spend on each chore. What is the correct number label for point A?
the correct number label for point A on the number line is [tex]\( \frac{1}{20} \).[/tex]
Let's break this down step by step:
1. Determine the time available for each chore: Amy has 1/4 hour to do 5 chores. To find out how much time she can spend on each chore, divide the total time available (1/4 hour) by the number of chores (5).
[tex]\( \frac{1}{4} \div 5 = \frac{1}{4} \times \frac{1}{5} = \frac{1}{20} \)[/tex] hour per chore.
2. Representing the fraction of an hour on a number line:The number line represents the whole hour. Since Amy has 1/20 hour per chore, each point on the number line represents 1/20 hour.
- Let's label point A as [tex]\( \frac{1}{20} \).[/tex]
So, the correct number label for point A on the number line is [tex]\( \frac{1}{20} \).[/tex]
Amy has 1/4 hour to complete 5 chores. To find out how much time she can spend on each chore, we divide 1/4 by 5, which gives us 1/20 hour per chore. On the number line representing an hour, each point represents 1/20 hour, so point A is labeled as 1/20. This means that Amy can spend 1/20 hour on each chore to complete all five chores within the 1/4 hour time frame she has.
complete question
Amy has 1/4 hour to do 5 chores. She plans to spend the same fraction of an hour on each chore. She wants to use the number line to help her determine what fraction of an hour she can spend on each chore. What is the correct number label for point A?
Which number is one million times larger than 8.7 x 104 ? A) 8.7 x 107 B) 8.7 x 108 C) 8.7 x 109 D) 8.7 x 1010
Answer:
D) [tex]8.7 * 10^{10}[/tex]
Step-by-step explanation:
The question is correctly written as below:
Which number is one million times larger than [tex]8.7 * 10^{4}[/tex]?
A) [tex]8.7 * 10^{7}[/tex]
B) [tex]8.7 * 10^{8}[/tex]
C) [tex]8.7 * 10^{9}[/tex]
D) [tex]8.7 * 10^{10}[/tex]
Consider [tex]8.7 x 10^{4}[/tex] to be number A and number B is one million times ([tex]10^{6}[/tex]) greater than number A
[tex]B = A * 10^{6}[/tex]
[tex]B = 8.7 * 10^{4} * 10^{6}[/tex]
[tex]B = 8.7 * 10^{4+6}[/tex]
[tex]B = 8.7 * 10^{10}[/tex]
D) [tex]8.7 * 10^{10}[/tex]
a company announced that 40% of those enrolled in its training to program will be sent to a new plant. if there are 60 people enrolled in the training program, how many will be sent to the new plant
There are 60 people enrolled in the program, then 24 people will be sent to the new plant.
What is the percentage?The percentage is defined as a ratio expressed as a fraction of 100.
To find the number of people sent to the new plant, we can multiply the percentage of people sent to the new plant by the total number of people enrolled in the training program. The percentage is given as a decimal by dividing the percentage by 100, so 40% is equal to 0.40.
The result is then multiplied by the number of people enrolled in the training program, which is 60.
40% of those enrolled in the training program will be sent to the new plant, which is 40/100 x 60 people = 24 people.
The final result is 24 people.
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An investment grows by 5% per year for 20 years. By what percent does it increase over the 20-year period?
An investment that grows by 5% per year for 20 years will increase by approximately 165.33% over the 20-year period. This calculation is based on the principle of compound interest.
Explanation:In the given scenario, the growth of the investment is calculated using the principle of
compound interest
- a concept in Mathematics. Compound interest takes into account the phenomenon that, over time, your investment will grow not only from the initial amount invested but also from any interest or earnings that have been accumulated. Therefore, the 5% growth rate per year is being applied to an increasingly larger amount because the interest is being compounded, i.e., you're earning 'interest on interest'.
To calculate the overall percent increase after 20 years, we use the formula for compound interest: A = P(1 + r/n)^(nt), where:
A is the final amount of money in the accountP is the principal amount (the starting amount)r is the annual interest rate (decimal)n is the number of times interest applied per time periodt is the number of time periods the money is invested forIn this case, we can assume that P = 1 (representing 100% of the initial investment), r = 0.05 (5% growth per year), n = 1 (interest is compounded once per year), and t = 20 years. Substituting these values in, we get A = 1(1 + 0.05/1)^(1*20) = 2.6533 approximately. Therefore, the investment increased by about 165.33% over the 20-year period.
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WILL MARK BRAINLIEST FOR THE BEST ANSWER!!!!!!!!!!!!
Find all complex solutions (real and non-real)
x^4+2x³+22x²+50x-75=0
You have just received an inheritance of $28,000 and would like to invest it into an account. The bank offers two investment plans, one for 4 years at 5.8% compounded annually and another for 3 years at 7.083% compounded annually. You want to make equal annual withdrawals from the account over the life time of the loan. Which investment will yield the highest return over the duration of the loan, given that the account will be zeroed out by the end of that period?
a.
3 year account; $32,056.89
c.
3 year account; $24,130.77
b.
4 year account; $42,742.52
d.
4 year account; $32,174.36
Answer:
D
Step-by-step explanation:
edge
10z when z = 0.8
answers
If the given function is F(x) = 10z. When z = 0.8, the value of the function is 8.
What is a function?It is defined as a special type of relationship, and they have a predefined domain and range according to the function every value in the domain is related to exactly one value in the range.
It is given that, the given function is F(x) = 10z.
In the given function z is the independent variable while the value of the function is a dependent variable because its value depends upon the value of z.
The variable is reliant on other measurable factors. As a result of manipulating the independent variable(s) in an experiment, these variables are anticipated to change.
The value of the function is obtained by putting the value of z in the given function,
F(x) = 10z
F(x) = 10×0.8
F(x) = 8
Thus, if the given function is F(x) = 10z. When z = 0.8, the value of the function is 8.
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Jay purchased tickets for a concert over the internet. To place an order, a handling charge of 5$ per ticket is charged. GST of 5% was also charged on ticket price and handling charges. If the total charge for 2 tickets is 201.16, what is the price per ticket.
Answer is 90.79 but I need an explanation on how they got that answer thanks
The price per ticket (before any additional charges) is approximately $90.79. Let's assume the price of each ticket before any additional charges (handling charge and GST) is "x" dollars.
Given that a handling charge of $5 is charged per ticket, the total handling charge for 2 tickets will be $5 * 2 = $10.
Now, let's calculate the total cost of the tickets after adding the handling charge:
Total cost of tickets (before GST) = 2x + 10
Now, the GST of 5% is charged on the ticket price and handling charges. The GST amount can be calculated as:
GST amount = 5% of (2x + 10) = 0.05 * (2x + 10) = 0.1x + 0.5
The total charge for 2 tickets, including the handling charge and GST, is given as $201.16. So, we can set up the equation:
Total cost of tickets (including GST) = Total cost of tickets (before GST) + GST amount
$201.16 = (2x + 10) + (0.1x + 0.5)
Now, solve for "x":
2x + 10 + 0.1x + 0.5 = 201.16
2.1x + 10.5 = 201.16
2.1x = 201.16 - 10.5
2.1x = 190.66
x = 190.66 / 2.1
x ≈ 90.79
So, the price per ticket (before any additional charges) is approximately $90.79.
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Find the total length of the curve with parametric equations x = sin(t), y = sin(2t), z = sin(3t)
The total length of the curve with parametric equations give is 23.5 units.
What are parametric equations?A parametric equation defines a group of quantities as functions of one or more independent variables called parameters.
Given that, a parametric equation, x = sin(t), y = sin(2t), z = sin(3t),
Finding the first derivative,
x' = cost
y' = 2cos2t
z' = 3cos3t
The parametric integral for length =
L = [tex]\int\limits^{t_2}_{t_1} \sqrt{ {x'^2+y'^2+z'^2} \, dx[/tex]
t₁ = 0, t₂ = 2π
T = [tex]\int\limits^{2\pi}_{0} \sqrt{ {(cost)^2+(2cos2t)^2+(3cos3t)^2} \, dt[/tex]
We will solve the integral by trapezoidal rule,
[tex]\int\limits^a_bf ({x}) \, dx = \frac{f(a)+f(b)}{2} (b-a)[/tex]
Therefore,
[tex]\int\limits^{2\pi}_{0} \sqrt{ {(cost)^2+(2cos2t)^2+(3cos3t)^2} \, dt[/tex]
= [tex]\frac{\sqrt{(cos2\pi)^2+(2cos2\pi)^2+(3+cos3\pi)^2} +\sqrt{(cos0)^2+(2cos0)^2+(3+cos0)^2}}2} (2\pi)[/tex]
= 23.5
Hence, the total length of the curve with parametric equations give is 23.5 units.
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Sixth-grade students at Cramon Junior High are not to have more than 12 hours of homework per week. They take one math course and two science courses in sixth grade. The amount of time required to do homework for a math course is x hours, and the time required to do homework for a science course is y hours.
How would this problem be represented in the form of an inequality?
Answer:
X+2y=12
Step-by-step explanation:
what is the unit rate of y=-2x
A rectangular storage container with an open top is to have a volume of 10 m3. the length of this base is twice the width. material for the base costs $10 per square meter. material for the sides costs $6 per square meter. find the cost of materials for the cheapest such container. (round your answer to the nearest cent.)
11<15+v
Solve the inequality for v