find the inverse of f informally. verify that f(f^-1(x))=x and f^-1(f(x))=x
f(x)= (x-1)/5
To find the inverse of a function, switch x and y and solve for y. The inverse of f(x) = (x-1)/5 is f^-1(x) = 5x + 1. Verify that f(f^-1(x)) = x and f^-1(f(x)) = x.
Explanation:To find the inverse of a function, we need to switch the roles of x and y and solve for y. In other words, we need to solve the equation y = (x-1)/5 for x. Let's do that:
Step 1: Replace y with x and x with y: x = (y-1)/5.
Step 2: Solve for y: Multiply both sides by 5: 5x = y - 1. Add 1 to both sides: 5x + 1 = y.
So, the inverse of f(x) = (x-1)/5 is f-1(x) = 5x + 1.
To verify that f(f-1(x)) = x, substitute f-1(x) into f(x) and simplify:
f(f-1(x)) = f(5x + 1) = ((5x + 1) - 1)/5 = 5x/5 = x.
Similarly, to verify that f-1(f(x)) = x, substitute f(x) into f-1(x) and simplify:
f-1(f(x)) = f-1((x-1)/5) = 5((x-1)/5) + 1 = x-1+1 = x.
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What is the value of x? If necessary, round your answer to two decimal places.
What is the exact value for the expression the square root of 56. − the square root of 14. + the square root of 126.? Simplify if possible.
the square root of 14.
4 the square root of 14.
2 the square root of 42.
8 the square root of 42.
...?
Answer:
B- [tex]\sqrt(56)-\sqrt(14)+\sqrt(126)=14.966[/tex]
Step-by-step explanation:
Given equation : square root of 56 − the square root of 14 + the square root of 126 or [tex]\sqrt(56)-\sqrt(14)+\sqrt(126)[/tex]
Solution of the equation :
[tex]\sqrt(56)-\sqrt(14)+\sqrt(126)[/tex]
= [tex]2\sqrt(14)-\sqrt(14)+3\sqrt(14)[/tex]
= [tex]\sqrt(14)(2-1+3)[/tex]
= [tex]4\sqrt(14)[/tex]
=14.966
Now,
square root of 14 =√14 =3.7414 the square root of 14 = 4√14 = 14.9662 the square root of 42= 2√42 = 12.9618 the square root of 42 = 8√42= 51.845Therefore, Option B is correct
Management for a chain of restaurants recorded the number of appetizers, X, ordered by tables dining. They observed that X had the following probability distribution.
Value of X
0
1
2
3 or more
Probability
0.60
0.35
0.04
0.01
Reference: Ref 9-7
The probability that a randomly chosen table orders at least one appetizer is
A. 0.35.
B. 0.39.
C. 0.40.
D. none of the above.
What segment is an altitude of triangle ABC?
segment AC
segment BC
segment BD
segment CD
...?
Jolie uses 2 tomatoes every day to prepare her salad. If n equals the number of tomatoes Jolie had before she made her salad and c equals the number of tomatoes left after the salad is made, which equation represents the number of tomatoes Jolie has after she made her salad?
n = 2 − c
c = n − 2
c = 2 − n
n = c − 2
please just give me the answer.
Answer:
d the answer is d
A piece of cloth is folded into a square with a side length measuring x inches. When it is unfolded, it has an area of x2 + 27x + 162 square inches.
Which could be the dimensions of the rectangular piece of cloth when it is unfolded? Remember, the area of a rectangle can be determined using the formula A = lw.
we need to factor the expression
[tex] x^{2} + 27x + 162 [/tex]
Equate to zero the expression
[tex] x^{2} + 27x + 162=0 [/tex]
Group terms that contain the same variable, and move the constant to the opposite side of the equation
[tex] x^{2} + 27x =-162 [/tex]
Complete the square. Remember to balance the equation by adding the same constants to each side
[tex] x^{2} + 27x +182.25=-162+182.25 [/tex]
[tex] x^{2} + 27x +182.25=20.25 [/tex]
Rewrite as perfect squares
[tex] (x+13.5)^{2}=20.25 [/tex]
[tex] (x+13.5)=(+/-)4.5 [/tex]
[tex] x1=-13.5+4.5= -9 [/tex]
[tex] x2=-13.5-4.5= -18 [/tex]
Hence
[tex] x^{2} + 27x + 162 =(x+9)*(x+18) [/tex]
Remember that the area of the rectangle is equal to
[tex] A=L*W [/tex]
therefore
the answer is
the dimensions of the rectangular piece of cloth could be
[tex] L=(x+18)\ inches\\ W=(x+9)\ inches [/tex]
Marietta drives her tow truck 312 miles in 8 hours. What is her average speed in miles per hour? Be sure to include the equation for speed in your answer.
The average speed in miles per hour is 39mi/hr
How to calculate Speed of a bodySpeed is the ratio of distance with respect to time. The formula for calculating speed is expressed as:
Speed = Distance/Time
Given the following
Distance = 312 miles
Time = 8 hours
Substitute
Speed = 312/8
Speed = 39mi/hr
Hence the average speed in miles per hour is 39mi/hr
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write 158/4 as a mixed number in simplest terms. Thank you.
Final answer:
The mixed number of 158/4 simplified is 39 1/2. A proportion comparing the ratios 1/48 and w/16 is solved by cross-multiplication resulting in w = 1/3 as the solution.
Explanation:
To write 158/4 as a mixed number in simplest terms, we first divide 158 by 4. When we do this division, 4 goes into 158, 39 times completely, since 4 * 39 = 156, and we get a remainder of 2. Therefore, the mixed number form of 158/4 is 39 2/4. To simplify this fraction further, we divide the numerator and the denominator by their greatest common divisor, which in this case is 2. Simplifying 2/4 by dividing both the numerator and denominator by 2, we get 1/2. Hence, the mixed number in its simplest terms is 39 1/2.
To set up a proportion using the given ratios, we start with the ratio 1/48 and set it equal to w/16, which can be represented as:
1/48 = w/16
When solving for w, we cross-multiply to get:
1 * 16 = 48 * w
Which simplifies to:
16 = 48w
Finally, to find the value of w, we divide both sides of the equation by 48, which gives us w = 16/48. Simplifying by dividing both numerator and denominator by the greatest common divisor, 16, we get:
w = 1/3
Hillary wants to go on the latin club trip to italy it cost 2730 for the trip the trip is 30 weeks away and she wants to make equal weekly payments how much money altogether does hillary need to pay at the end of week eight
Which term describes a rate in which the second quantity is 1?
unit rate
percent
proportion
ratio
The function p(t) represents the value of a new car t years after it was manufactured.
p(t)=24,525(0.94)^t
What does the value 0.94 represent in this situation?
The value of the car increases by 94% each year.
The value of the car decreases by 94% each year.
The value of the car is 0.94 times the value of the car the previous year.
The initial value of the car is 0.94.
Answer:
The value of the car is 0.94 times the value of the car the previous year.
The school that Scott goes to is selling tickets to a fall musical. On the first day of ticket sales the school sold 1 senior citizens tickets and 12 students tickets for a total of $130. The school took in $80 on the second day by selling 5 senior citizens tickets and 3 student tickets. Find the price of a senior citizens ticket and the price of a student tickets
Which transformations map the strip pattern onto itself
Answer: a horizontal translation and a 180 rotation
Step-by-step explanation:
what is the answer to this 10-4b
The gross weight of a package and its contents is 11 pounds 5 ounces . If the packaging weighs 1 pound 15 ounces , what's the net weight of the contents
On Saturday, a local hamburger shop sold a combined total of 464 hamburgers and cheeseburgers. The number of cheeseburgers sold was three times the number of hamburgers sold. How many hamburgers were sold on Saturday?
The number of hamburgers will be 348.
Expression in maths is defined as the collection of numbers variables and functions by using signs like addition, subtraction, multiplication, and division.
Numbers (constants), variables, operations, functions, brackets, punctuation, and grouping can all be represented by mathematical symbols, which can also be used to indicate the logical syntax's order of operations and other features.
Given that On Saturday, a local hamburger shop sold a combined total of 464 hamburgers and cheeseburgers. The number of cheeseburgers sold was three times the number of hamburgers sold.
The number of hamburgers will be calculated as below:-
Cheeseburgers= x
Hamburgers= 3x
3x+x=464
4x=464
x=116
Substitute the value of x in expression.
Hamburgers= 3x
Hamburgers= 3 x 116 = 348
Therefore, the number of hamburgers will be 348.
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The local hamburger shop sold 116 hamburgers on Saturday. We determined this by setting up an equation where the number of hamburgers (H) plus three times the hamburgers (3H) equaled the total sales of 464 burgers, and then solving for H.
Explanation:The question asks us to solve a basic algebra problem involving hamburgers and cheeseburgers. Let's define the number of hamburgers sold as H. Since the number of cheeseburgers sold was three times the number of hamburgers sold, we can represent the number of cheeseburgers as 3H. The total number of burgers sold was 464. Now, we can write down the equation:
H + 3H = 464
This simplifies to:
4H = 464
Now, we divide both sides by 4 to find the value of H:
H = 464 / 4
H = 116
So, the local hamburger shop sold 116 hamburgers on Saturday.
The Smiths plan to recarpet their family room, which measures 15 ft. by 20 ft. How many square yards of carpet are needed?
Answer: [tex]33\frac{1}{3}[/tex] square yards of carpet is needed.
Step-by-step explanation:
Given: The dimensions for the carpet is 15 ft. by 20 ft.
We know that area ( in square units ) of a rectangle is given by :-
[tex]Area=length\times width[/tex]
Therefore, area of the required carpet is given by :-
[tex]Area=20\times15=300\text{ square feet}[/tex]
We know that 1 feet =1/3 yard
Now,
[tex]300\text{ square feet}=300\times\frac{1}{3}\times\frac{1}{3}\\=\frac{100}{3} \text{square yards}=33\frac{1}{3}\text{ square yards}[/tex]
Hence, [tex]33\frac{1}{3}[/tex] square yards of carpet is needed.
name the property used in each equation. then find the value of n.
2.) 1·n=8
3.) 28·n=0
4.) 0+n=22
5.) 1/4·n=1
integrate 1 / sqrt(25 - 16x^2) dx
To integrate 1 / sqrt(25 - 16x^2) dx, we use trigonometric substitution with x = (5/4)sinθ. After substitution, integration, and back-substitution, the result is arcsin(4x/5) + C, where C is the constant of integration.
Explanation:The integral to solve is ∑ (1 / sqrt(25 - 16x^2)) dx, which resembles the inverse trigonometric function, specifically the arcsine function. To solve this integral, we can use a trigonometric substitution. Since the integral involves a square root of a difference of squares, we can substitute x with (5/4)sinθ, which simplifies the integral considerably.
After substitution, we can then use the derivative of sinθ, which is cosθ, in order to find the full differential dx to replace in our integral. The resulting integral in terms of θ can be integrated using standard techniques, yielding an arcsine function as the antiderivative. After integrating, we must then convert back from θ to x to express the solution to the original integral in the original variable.
The answer to the integral ∑ (1 / sqrt(25 - 16x^2)) dx is arcsin(4x/5) + C, where C is the constant of integration. The substitution step is crucial in solving this type of problem and is a commonly used method in calculus.
Solve the system using elimination.
5x + 4y = 12
3x – 3y = 18
Final answer:
To solve the system of equations using elimination, multiply one equation by a constant so that the coefficients of one variable will be the same in both equations. Then add or subtract the equations to eliminate that variable. Finally, solve for the remaining variable and substitute back to find the complete solution.
Explanation:
To solve the system of equations using elimination, we need to eliminate one variable by adding or subtracting the two equations. In this case, we can eliminate the variable y by multiplying the first equation by 3 and the second equation by 4. This will give us:
15x + 12y = 36
12x - 12y = 72
Next, we can add the two equations together to eliminate y:
27x = 108
Dividing both sides by 27, we find that x = 4.
Substituting this value of x back into one of the original equations, we can solve for y:
5x + 4y = 12
5(4) + 4y = 12
20 + 4y = 12
Subtracting 20 from both sides, we get 4y = -8.
Dividing both sides by 4, we find that y = -2.
Therefore, the solution to the system of equations is x = 4 and y = -2.
Find the exact area of a circle having the given circumference.
4(sqrt3 PI ) ...?
Answer:
12(pi)
Step-by-step explanation:
graph the fraction 1/2 on the number line.
the number of equally sized spaces needed between 0 and 1 is ?
Carol buys a reference book that originally cost $87.50 for 15% off. She pays a sales tax of 4.5%. What is the total cost of the book?
Answer:
$77.72
Step-by-step explanation:
This is the total cost of the book with tax.
What are the coordinates of the vertex for f(x) = 2x^2 + 4x + 9?
(−2, 4)
(−1, 7)
(1, 15)
(2, 9)
What are the roots of the equation 2x2 + 9x + 6 = 0?
Find the domain and range of the relation and determine whether it is a function.
Domain: positive integers; range: positive integers; no, it is not a function.
Domain: x > 3; range: y > 0; yes, it is a function.
Domain: all real numbers; range: all real numbers; yes, it is a function.
Domain: x ³ 0; range: y > 3; no, it is not a function.
Answer:
y>0
Step-by-step explanation:
in the fraction 3/5 which statement best describes the denominator
The increase in a person’s body temperature T(t), above 98.6ºF, can be modeled by the function T(t)=(4t)/(t^2 +1), where t represents time elapsed. What is the meaning of the horizontal asymptote for this function? The horizontal asymptote of y = 0 means that the person’s temperature will approach 98.6ºF as time elapses. The horizontal asymptote of y = 0 means that the person’s temperature will approach 0ºF as time elapses. The horizontal asymptote of y = 4 means that the person’s temperature will approach 102.6ºF as time elapses. The horizontal asymptote of y = 4 means that the person’s temperature will approach 4ºF as time elapses.
The person's body temperature will approach 98.6ºF as time lapses. Option A is correct.
How to calculate the horizontal asymptotes of a function?A horizontal asymptote for a function is an imaginary line that is not part of the graph and lies along the x-axis of the graph either to the left or right.
The horizontal asymptote of the given function can be described as the nature of the function T(f) as t approaches infinity.
This is the point where the line y=0 which means that the person's body temperature will approach 0ºF above 98.6ºF as t goes to infinity.
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Samuel drove 28 miles in 32 minutes. Beverly drove 42 miles in 48 minutes.
Are the ratios of miles to minutes for Samuel and Beverly proportional?
Answer:
Yes. Ratios of miles to minutes for Samuel and Beverly are proportional
Step-by-step explanation:
Samuel's speed can be calculated as following:
[tex]28/32=0,875[/tex]
0,875 miles per minute.
On the other hand Beverly's speed can be calculated as following:
[tex]42/48=0,875[/tex]
0,875 miles per minute
If we convert the speed to mile per hour:
[tex]0,875*60=52,5[/tex]
It can be easily seen that both Samuel and Beverly drove at the same speed at 52,5 miles per hour.