We take the inverses of the function and get the results as follows:
[tex]f(1)=1+1=2\\f^{-1}(1)=\sqrt{1-1}=0\\f^{-1}(f(1))=\frac{\sqrt{\sqrt{4(1)+1} -3}}{\sqrt{2} } \\f(f^{-1} (1))=1[/tex], [tex]f(2)=4+1=5\\f^{-1}(2)=\sqrt{2-1}=1\\f^{-1}(f(2))=\frac{\sqrt{\sqrt{4(2)+1} -3}}{\sqrt{2} } \\f(f^{-1} (2))=2[/tex], [tex]f(3)=9+1=\sqrt{10} \\f^{-1}(3)=\sqrt{3-1}=\sqrt{2} \\f^{-1}(f(3))=\frac{\sqrt{\sqrt{4(3)+1} -3}}{\sqrt{2} } \\f(f^{-1} (3))=3[/tex]
What are the inverses of a function?
A function that can change into another function is referred to as being inverse or anti. In other words, the inverse of any function "f" will take y to x if it takes x to y in the first place. The inverse function is indicated by F^-1 if the function is denoted by F.
Given C={1,2,3}, and
[tex]f(x)=x^{2} +1\\ f^{-1}(x)=\sqrt{x-1} \\f^{-1}(f(x))=\frac{\sqrt{\sqrt{4(x)+1} -3}}{\sqrt{2} } \\f(f^{-1} (x))=x[/tex]
Now,
For C=1,
[tex]f(1)=1+1=2\\f^{-1}(1)=\sqrt{1-1}=0\\f^{-1}(f(1))=\frac{\sqrt{\sqrt{4(1)+1} -3}}{\sqrt{2} } \\f(f^{-1} (1))=1[/tex]
For C=2
[tex]f(2)=4+1=5\\f^{-1}(2)=\sqrt{2-1}=1\\f^{-1}(f(2))=\frac{\sqrt{\sqrt{4(2)+1} -3}}{\sqrt{2} } \\f(f^{-1} (2))=2[/tex]
For C=3
[tex]f(3)=9+1=\sqrt{10} \\f^{-1}(3)=\sqrt{3-1}=\sqrt{2} \\f^{-1}(f(3))=\frac{\sqrt{\sqrt{4(3)+1} -3}}{\sqrt{2} } \\f(f^{-1} (3))=3[/tex]
Hence, the solution is provided.
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If a polynomial f(x) has a remainder of 4 when divided by x-5, what is f(5)?
The value of f(5) When a polynomial f(x) is divided by x -5 is 4
What is a polynomial?Polynomials are algebraic expressions that consist of variables and coefficients. Variables are also sometimes called indeterminates. We can perform arithmetic operations such as addition, subtraction, multiplication, and also positive integer exponents for polynomial expressions but not division by variable.
If f(x) has a remainder of 4 when divided by x - 5
Then it means that if f(x -5) = f(0)
x -5 = 0
x = 5
so if you substitute 5 into f(x) which is f(5) it will give 4
so therefore f(5) = 4
In conclusion f(5) is same as 4
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Devaughn's age is three times Sydney's age. The sum of their ages is 116. What is Sydney's age?
Hello,
I hope you and your family are doing well!
Let's call Sydney's age x and Devaughn's age y. We know that y = 3x and x + y = 116.
Substituting the first equation into the second equation gives:
3x + x = 116
4x = 116
x = 29
So Sydney's age is 29.
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Breaking strengths of a random sample of 28 molded plastic housing were measured, and a sample standard deviation of S=19.52 was obtained. A confidence interval (439.23, 460.77) for the average strength of molded plastic housings were constructed from these results. What is the confidence level of this confidence interval?
The confidence interval of this confidence level as calculate from the given data is 3.68 and 99% respectively.
Number of random sample (n)=28
Sample Standard deviation (S)=19.52
Confidence interval = 439.23 - 460.77 = 21.54
t = s/root of n
=19.52 /5.29
=3.68
v = n -1 = 27
now, confidence level
= 1 - alpha
= 99%
Confidence Interval, put simply, is a range within which we are confident that the genuine value exists. The probability that a confidence interval will contain the real parameter value depends on the confidence level that is chosen for the interval.
The phrase "Confidence Interval" refers to the range of values that is typically used to handle population-based data, extracting precise, important information with a given level of confidence.
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Write a system of equations to describe the situation below, solve using
substitution, and fill in the blanks.
Tucker is collecting pledges for a walk-a-thon. His mother has pledged a flat
donation of $5, and his grandmother has pledged $5 per mile. If Tucker walks
a certain distance, the two donors will end up owing the same amount. How
much will each donor owe? What is that distance?
Tucker's mother and grandmother will each owe $
miles.
if he walks
The distance for Tucker to walk is 1 mile, only then both the donors would have the same amount. This was found using linear equations.
What are linear equations?
A linear equation is one in which the variable's maximum power is consistently 1. A one-degree equation is another name for it. A linear equation with one variable has the formula Ax + B = 0, which is its standard form. x is a variable, A is a coefficient, and B is constant in this situation. A linear equation with two variables has the formula Ax + By = C as its standard form. Here, the variables are x and y, the coefficients are A and B, and the constant is C.
Since it's given that Tucker's mother will give flat $5. So let that be the RHS of the equation. Now his grandmother will give him $5 for every mile he walks.
Let that be 5x.
Now the equation would be: 5x=5
Which results in x=1.
Therefore at 1 mile, both of the donors' amounts would be the same.
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Rewrite the following function in factored form. f(x)=x2+6x+8
Answer:
f(x) = (x+2)(x+4)
Step-by-step explanation:
4 + 2 = 6
4 * 2 = 8
f(x) = (x+2)(x+4)
Solve for x. Round to the nearest tenth.
The value of x for the triangle is equal to 17.32.
What is the Pythagorean theorem?Pythagorean theorem states that in the right angle triangle the hypotenuse square is equal to the square of the sum of the other two sides. According to this statement, the areas of the squares on the other two sides add up to the size of the square whose side is the hypotenuse.
Calculate the perpendicular side for both triangles by using the Pythagorean theorem.
H² = P² + B²
Here, H is the hypotenuse, P is perpendicular and B is the base.
For the first triangle, the perpendicular is calculated as,
P² = x² - 10²......................................................1
For the second triangle, the perpendicular will be,
P² = ( 30² - x² ) - 20² .......................................2
Equate the equation 1 and 2 and solve,
x² - 10² = ( 30² - x² ) - 20²
x² - 100 = 900 - x² - 400
2x² = 900 - 400 + 100
2x² = 600
x√ = 300
x =√300
x = 17.32
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3. The diameter of an atom is 0.0000000001 inches. Which of these is an appropriate
estimate for the diameter of an atom?
1×10-10
A.
B.
1×10-9
C. 1×10¹0
D. 1×1019
in.
in.
in.
in.
Answer:
a. [tex]1 * 10^{-10}[/tex]
Explanation:
A converted to standard notation is the same number.
After Becca's half-marathon, her pedometer showed 23, 580 steps, while Alex recorded
2.42 x 104 steps after the same race. What is the combined amount of their steps?
A
4.778 x 103 ; 4,778
B
2.3822 x 104 ; 23,822
C
4.778 x 104 ; 47, 780
D
4.808 x 104 ; 48,080
Answer:
Below
Step-by-step explanation:
2.42 x 10^4 = 24200
24200
+ 23580
47 780 or 4.778 x 10^4
After going to your favorite restaurant for dinner and buying food for everybody the bill came to $145. You decide to leave 30% tip and the tax came to 10%.
How much was your total bill?
Answer:
203 dollars
Step-by-step explanation:
100 percent = 145
10 percent = 14.5
30 percent = 43.5
43.5 + 14.5 + 145 = 203
The diameter of a speck of dust is about 0.00002 inches. Which of these is an appropriate estimate for the diameter of the speck of dust?
A
2 x 10-5 in.
2 x 10-4 in.
2 x 104 in.
2 x 105 in.
B
с
D
The appropriate estimate for the diameter of the speck of dust is 2 × 10⁻⁵ inches. So the answer is option A, 2 × 10⁻⁵ in.
What is Scientific Notation:In scientific notation, numbers are written with the base 10 much like in decimal notation. Scientific notation is the term used when a number is expressed as the product of two numbers.
Where the first number is more than or equal to one but less than ten and the second number is the power of ten expressed in exponential form.
Convert Decimal to Scientific Notation:Step - 1 Move the decimal point up to the first number that is equal to 1 but less than 10.
Step - 2 Count the number of places that the point moved.
Step - 3 Now Write the resultant number as a product with a power of 10.
Here we have
Diameter of a speck of duct = 0.00002 inches
Convert the given decimal into a scientific notation
As we learn above
Move the decimal point up to a whole number toward the right
=> 0.00002 = 2. = 2
Count the number of zeros or places moved
=> n = 5
Now write the number as a power of 10, Now write the above whole number part as the product of 10
=> 2 × 10⁻⁵ [ ∵ The power is moved right we need to use negative ]
Therefore,
The appropriate estimate for the diameter of the speck of dust is 2 × 10⁻⁵ inches. So the answer is option A, 2 × 10⁻⁵ in.
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Maximize: Profit = 20x + 30y
Constraints:
x + y <12
x + 2y<12
2x + y<12
x>0
y>0
Answer:
x> -10
Step-by-step explanation:
Complete the square by adding the correct missing term on the left, then factor as indicated:
x²+x+_____________=(_____)²
The complete sentence will be;
⇒ x² + x + 1/4 = ( x + 1/2)²
What is an expression?
Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.
Given that;
The expression is,
⇒ x² + x + __ = (_____)²
Now,
Since, The expression is,
⇒ x² + x + __ = (_____)²
Hence, We can add the number to form the complete square as;
⇒ x² + x + 1/4 = x² + x + (1/2)²
= ( x + 1/2)²
Thus, The complete sentence will be;
⇒ x² + x + 1/4 = ( x + 1/2)²
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In a lottery game, a player picks six numbers from 1 to 28. If the player matches all six numbers, they win 50,000 dollars. Otherwise, they lose $1.
What is the expected value of this game?
The expected value of this game will be 1,784.75.
What is the expected value?In parameter estimation, the expected value is an application of the weighted sum. Informally, the expected value is the simple average of a considerable number of individually determined outcomes of a randomly picked variable.
The expected value is given below.
E(x) = np
Where n is the number of samples and p is the probability.
In a lottery game, a player picks six numbers from 1 to 28. Assuming the player matches each of the six numbers, they win 50,000 bucks. In any case, they lose $1.
Then the probability of winning (p) and losing (q) is given as,
p = 1 / 28
q = 27 / 28
Then the expected value is given as,
E(x) = 50,000 (1 / 28) - 1 (27 / 28)
E(x) = 1785.714 - 0.964
E(x) = 1,784.75
The expected value of this game will be 1,784.75.
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In a video game, Clare scored 50% more points than Tyler. If c is the number of points that Clare scored and t is the number of points Tyler scored, which equations, are correct?
Answer:
c= 1.5t, c=t+0.5t and c=(1+0.5)t
Step-by-step explanation:
Given that, Clare scored 50% more points than Tyler, c is the number of points that Clare scored and t is the number of points that Tyler scored,According to question,c = 150% of tc = 1.5tHence, the correct options are c = 1.5t, c = t+0.5t and c = (1+0.5)t
WORD PROBLEM:
When the square of a certain number is added to the number, the resu;t is the same as when 48 is added to three times the number. Find the number.
Answer:
Step-by-step explanation:
Let x be the number.
The square of the number is x2
Increased by twice the number means add 2x.
So your equation is :
X2 +2x = 48.
Can you take it from here?
The asked number is 12.
What is an equation?An equation is a mathematical statement that shows that two mathematical expressions are equal.
Given that, When the square of a certain number is added to the number, the result is the same as when 48 is added to three times the number.
Let the number be x,
x²+x = 48X3 + x
x² = 144
x = 12
Hence, The asked number is 12.
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Given vector v = (1, -1), find -4v.
Answer:
<-4,4>
Step-by-step explanation:
-4 * <1,-1> = <-4 * 1, -4 * -1> = <-4, 4>
X times a number is greater than or equal to 6
The equation, which represents the given condition x >= 6.2.
What is the inequality equation?
Equations and inequalities are mathematical sentences formed by relating two expressions to each other. In an equation, the two expressions are supposed to be equal and shown by the symbol =. Whereas in inequality, the two expressions are not necessarily equal and are indicated by the symbols: >, <, ≤, or ≥.
We have,
Six more than 5 times a number x is greater than or equal to 6.
So,
5x >= 31
5x >= 31/5
x >= 6.2
Hence, the equation, which represents the given condition x >= 6.2.
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Complete equation:
Six more than 5 times a number x is greater than or equal to 6.
Suppose S = {1, 2, 3}, with P({1}) = 1/2 and P({1, 2}) = 2/3. What must P({2}) be?
The value of P({2}) for the given Finite set is found as P({2}) = 1/6.
Explain the meaning of the term finite set?Sets with a bounded number of participants are referred to as finite sets. Due to their ability to be numbered, finite sets also are known as countable sets. If the members of the elements of all this set are finite, then the operation will run from out elements to list. Finite set illustrations include P = { 0, 3, 6, 9, …, 99}For the stated question;
S = {1, 2, 3},P({1}) = 1/2 and P({1, 2}) = 2/3.P({2}) will contain only element with is not present in set P({1}) counting from P({1, 2}).
Thus,
P({2}) = P({1, 2}) - P({1})
P({2}) = 2/3 - 1/2
P({2}) = 1/6
Thus, the value of P({2}) for the given Finite set is found as P({2}) = 1/6.
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Question 5 (1 point)
Dev has a monthly food budget of $182. He maps the amount of money, x, he spends each month to the number of food items he buys.
What are the constrains on the domain?
O a
domain:x>182
Ob
domain: x20
Oc domain: x ≤182
Od domain: 0≤x≤182
For the function in the given case the constraint on domain can be written as x ≥ 182. The correct option is (a).
What is a function?A function y = f(x) is a one to one relationship between two sets X and Y where the set X is called the domain and Y the range of function f(x) ans x ∈ X and y ∈ Y.
Suppose the amount of money spent be x.
And, the number of food items be y.
Now, as per the question the function can be written as follows,
y = x/182
As the number of food items can only be positive integers, the values of x must be the multiples of 182.
Thus, the expression for the domain can be written as x ≥ 182.
Hence, the constraint on the domain is given as x ≥ 182.
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(Help)
What is the y-incept of the following equation ?
X=3
Is y 0? Or am I wrong?
Answer: There is no y-intercept on this equations
Step-by-step explanation:
so yes you were wrong
What is the value of p in the equation p3=−0.027 ?
Answer:I hope this helpful mark me brainlist if correct then no if wrong
Step-by-step explanation:
P times 3 =-0.0027
Use the commutative property to reorder the terms
3p=-0.0027
Divide both sides of the equation by 3
p=-0.009 or p=-9/1000
In a recent year 365 days a hospital had 5814 births. find the mean number of births per day
Answer:
ANS:- 15.9
Step-by-step explanation:
5814/365 = 15.9
Factor completely 8x^3+216
Step-by-step explanation:
8x³ + 216
(2 * 2 * 2 * x * x * x) + (2 * 3 * 2 * 3 * 2 * 3)
A hexagon has a perimeter of 34 2/3 inches. What is the length of one side.
Answer:
Length of each side of regular hexagon = 19 inches
Step-by-step explanation:
What is a regular hexagon ?When the length of all sides is equal and measure of all angles is equal . it is known as Regular hexagon. In a regular hexagon, each inside angle is 120 degrees.
Perimeter of Regular hexagon = 6a
where a is each side of regular hexagon
According to the given information
Perimeter of regular hexagon = [tex]\frac{342}{3}[/tex]
= 114 inches
Perimeter of regular hexagon = 6a
114 = 6a
19 = a
Length of each side of regular hexagon = 19 inches
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"a. What was the population of the state in 2000 ?
b. When will the population of the state reach 28.3 million?
a. In 2000 , the population of the state was million."
In 2000, the population of the state was 23.1 million, After 13 years , the population of the state reach 28.3 million.
What is population?
The term "population" refers to all citizens who are present in a nation or who are temporarily absent from it, as well as all aliens who have made a nation their permanent home. The number of residents in a given area is depicted by this indicator. The annual changes in population brought on by births, deaths, and net migration are measured as growth rates.
Given:
[tex]A = 23.1e^0^.^0^1^5^2^t[/tex] models the population of US
where A is in million
t is years after 2000.
(a) We have to find the population in 2000.
In 2000, t = 0
Plug t = 0 in the above equation
[tex]A(0) = 23.1e^0^.^0^1^5^2^(^0^)\\A = 23.1[/tex]
Hence, in 2000 the population is 23.1 million.
(b) To find when the population of the state reach 28.3 million.
Plug A = 28.3 in above equation and solve for t.
[tex]28.3 = 23.1e^0^.^0^1^5^2^t\\e^0.0152t = \frac{28.3}{23.1} \\t*ln(0.0152) = ln( \frac{28.3}{23.1})\\t = 13.35718\\t = 13[/tex]
Hence, after 13 years the population of the state reach 28.3 million.
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-6x+3y=24 in slope intercept form
Answer:
y=2x+8
Step-by-step explanation:
-6x+3y=24
-6x+6x+3y=24+6x ==> isolate the y variable
3y=6x+24 ==> simplify
(3y=6x+24)/3
y=2x+8 <== y=mx+b <== slope intercept form
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PLS HELP! 70 POINTS!!
Mr. Mudd gives each of his children $2000 to invest as part of a friendly family competition. The competition will last 10 years. The rules of the competition are simple. Each child can split up his or her $2000 into as many separate investments as they please. The children are encouraged to do their research on types of investments. The initial investments made may not be changed at any point during the 10 years; no money may be added and no money may be moved. Whichever child has made the most money after 10 years will be awarded an additional $10,000.
The table shows the investment made by each of Mr. Mudd’s children, and what happened to their investment over the decade long friendly competition.
Child Performance of investments over the course of the competition
Albert $1000 earned 1.3% annual interest compounded monthly
$500 lost 3% over the course of the 10 years
$500 grew compounded continuously at rate of 0.7% annually
Marie $1500 earned 1.3% annual interest compounded quarterly
$500 gained 3% over the course of 10 years
Hans $2000 grew compounded continuously at rate of 0.8% annually
Max $1000 decreased in value exponentially at a rate of 0.6% annually
$1000 earned 1.9% annual interest compounded biannually (twice a year)
Make sure to show your work
1. What is the balance of Albert’s $2000 after 10 years?
2. What is the balance of Marie’s $2000 after 10 years?
3. What is the balance of Hans’ $2000 after 10 years?
4. What is the balance of Max’s $2000 after 10 years?
5. Who is $10,000 richer at the end of the competition?
Answer:
Albert = $2159.07; Marie = $2244.99; Hans = $2188.35; Max = $2147.40
Marie is $10 000 richer
Step-by-step explanation:
Albert
(a) $1000 at 1.2 % compounded monthly
A = P (1 + r/n )^nt
A = 1000(1 + 0.001)¹²⁰ = $1127.43
(b) $500 losing 2%
0.98 × 500 = $490
(c) $500 compounded continuously at 0.8%
A = Pe^rt
= 500e^0.008 * 10
= $541.64
(d) Balance
Total = 1127.43 + 490.00+ 541.64 = $2159.07
Marie
(a) 1500 at 1.4 % compounded quarterly
A = 1500(1 + 0.0035)⁴⁰ = $1724.99
(b) $500 gaining 4 %
1.04 × 500 = $520.00
(c) Balance
Total = 1724.99 + 520.00 = $2244.99
Hans
$2000 compounded continuously at 0.9 %
A = 2000e^0.009 * 10
= $ 2188.35
Max
(a) $1000 decreasing exponentially at 0.5 % annually
A = 1000(1 - 0.005)¹⁰= $951.11
(b) $1000 at 1.8 % compounded biannually
A = 1000(1 + 0.009)²⁰ = $1196.29
(c) Balance
Total = 951.11 + 1196.29 = $2147.40
Marie is $10,000 richer at the end of the competition.
Hope this helps!!! :)
5400 dollars is placed in an account with an annual interest rate of 6.75%. To the nearest tenth of a year, how long will it take for the account value to reach 11200 dollars?
The time needed for the account value to reach 11200 dollars is approximately 15.9 years.
What is the time required to reach the accrued amount?Simple interest formula is expressed as;
A = P( 1 + rt )
Where A is accrued amount, P is principal, r is interest rate and t is time.
Given the data in the question;
Accrued amount A = $11200 Principal P = $5400Interest I = 6.75%Time r = ?Plug the given values into the above formula and solve for time t.
A = P( 1 + rt )
A/P = 1 + rt
( A/P - 1 ) = rt
t = ( A/P - 1 ) / r
t = ( $11200/$5400 - 1 ) / 6.75%
t = ( $11200/$5400 - 1 ) / 6.75/100
t = ( 56/27 - 1 ) / 0.0675
t = ( 29/27 ) / 0.0675
t = 15.9 years.
Therefore, the required time to reach the accrued amount is 15.9 years.
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Which of the following points is a solution to the system of inequalities?
(A) (-4,4)
(B) (2,6)
(C) (8,-6)
(D) (6,3)
Answer:
Step-by-step explanation:
Answer:
A. is the answer
Step-by-step explanation:
A. Since there is one value of y for every value of x
B. is not a function x = − 4 produces y = 4 and y = 3
C. is not a function x = 6 produces y = 6 and y = 3
D. is not a function x = 6 produces y = 3 and y = 5
What is the area of a semicircle with a diameter of 20?
Answer:
[tex]\huge\boxed{\sf 157.1 \ units^2}[/tex]
Step-by-step explanation:
Diameter = 20
Radius = Diameter/2 = 20/2 = 10
Area of semicircle:[tex]\displaystyle =\frac{\pi r^2}{2} \\\\r = 10\\\\= \frac{\pi (10)^2}{2} \\\\= \frac{100 \pi}{2} \\\\= \frac{314.2}{2} \\\\= 157.1 \ units^2\\\\\rule[225]{225}{2}[/tex]