Answers
Given:
The function is
\(f(x)=-\dfrac{1}{2}\sqrt{x+3},x\geq -3\)
To find:
The inverse of the given function.
Solution:
We have,
\(f(x)=-\dfrac{1}{2}\sqrt{x+3}\)
Put f(x)=y.
\(y=-\dfrac{1}{2}\sqrt{x+3}\)
Interchange x and y.
\(x=-\dfrac{1}{2}\sqrt{y+3}\)
Isolate the variable y.
\(-2x=\sqrt{y+3}\)
\((-2x)^2=y+3\)
\(4x^2-3=y\)
\(y=4x^2-3\)
Putting \(y=f^{-1}(x)\), we get
\(f^{-1}(x)=4x^2-3\)
For \(x\geq -3\),
\(x+3\geq 0\)
\(\sqrt{x+3}\geq 0\)
\(-\dfrac{1}{2}\sqrt{x+3}\leq 0\)
\(f(x)\leq 0\)
It means for function \(f(x)\leq 0\) and for inverse function \(x\leq 0\) because the range of the function is the domain of inverse function.
Therefore, \(f^{-1}(x)=4x^2-3\) for \(x\leq 0\).
Related Questions
how many integers between 2023 and 5757 have 12, 20, and 28 as factors
Answers
Answer:
9 integers between 2023 and 5757 that have 12, 20, and 28 as factors.
Step-by-step explanation:
An integer that has 12, 20, and 28 as factors must be divisible by the least common multiple (LCM) of these numbers. The LCM of 12, 20, and 28 is 420. So we need to find the number of integers between 2023 and 5757 that are divisible by 420.
The first integer greater than or equal to 2023 that is divisible by 420 is 5 * 420 = 2100. The last integer less than or equal to 5757 that is divisible by 420 is 13 * 420 = 5460. So the integers between 2023 and 5757 that are divisible by 420 are 2100, 2520, ..., 5460. This is an arithmetic sequence with a common difference of 420.
The number of terms in this sequence can be found using the formula for the nth term of an arithmetic sequence: an = a1 + (n - 1)d, where an is the nth term, a1 is the first term, d is the common difference, and n is the number of terms. Substituting the values for this sequence, we get:
5460 = 2100 + (n - 1)420 3360 = (n - 1)420 n - 1 = 8 n = 9
So there are 9 integers between 2023 and 5757 that have 12, 20, and 28 as factors.
Hey, I need help with questions 1 and 2, if anyone can help, please? thanks!
Answers
The correct works are:
\(Blue(s + h) = 2s^2 + 4sh + 2h^2 + 3\).\(\frac{Blue(s + h) - Blue(s)}{h} = 4s + 2h\)Function NotationThe function is given as:
\(Blue(s) = 2s^2 + 3\)
The interpretation when Steven is asked to calculate Blue(s + h) is that:
Steven is asked to find the output of the function Blue, when the input is s + h
So, we have:
\(Blue(s + h) = 2(s + h)^2 + 3\)
Evaluate the exponent
\(Blue(s + h) = 2(s^2 + 2sh + h^2) + 3\)
Expand the bracket
\(Blue(s + h) = 2s^2 + 4sh + 2h^2 + 3\)
So, the correct work is:
\(Blue(s + h) = 2s^2 + 4sh + 2h^2 + 3\)
Simplifying Difference QuotientIn (a), we have:
\(Blue(s + h) = 2s^2 + 4sh + 2h^2 + 3\)
\(Blue(s) = 2s^2 + 3\)
The difference quotient is represented as:
\(\frac{f(x + h) - f(x)}{h}\)
So, we have:
\(\frac{Blue(s + h) - Blue(s)}{h} = \frac{2s^2 + 4sh + 2h^2 + 3 - 2s^2 - 3}{h}\)
Evaluate the like terms
\(\frac{Blue(s + h) - Blue(s)}{h} = \frac{4sh + 2h^2}{h}\)
Evaluate the quotient
\(\frac{Blue(s + h) - Blue(s)}{h} = 4s + 2h\)
Hence, the correct work is:
\(\frac{Blue(s + h) - Blue(s)}{h} = 4s + 2h\)
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John wishes to choose a combination of two types of cereals for breakfast - Cereal A and Cereal B. A small box (one serving) of Cereal A costs $0.50 and contains 10 units of vitamins, 5 units of minerals, and 15 calories. A small box (one serving) of Cereal B costs $0.40 and contains 5 units of vitamins, 10 units of minerals, and 15 calories. John wants to buy enough boxes to have at least 500 units of vitamins, 600 units of minerals, and 1200 calories. How many boxes of each cereal should he buy to minimize his cost?
Answers
Let's assume that John buys x boxes of Cereal A and y boxes of Cereal B. Then, we can write the following system of inequalities based on the nutrient and calorie requirements:
10x + 5y ≥ 500 (minimum 500 units of vitamins)
5x + 10y ≥ 600 (minimum 600 units of minerals)
15x + 15y ≥ 1200 (minimum 1200 calories)
We want to minimize the cost, which is given by:
0.5x + 0.4y
This is a linear programming problem, which we can solve using a graphical method. First, we can rewrite the inequalities as equations:
10x + 5y = 500
5x + 10y = 600
15x + 15y = 1200
Then, we can plot these lines on a graph and shade the feasible region (i.e., the region that satisfies all three inequalities). The feasible region is the area below the lines and to the right of the y-axis.
Next, we can calculate the value of the cost function at each corner point of the feasible region:
Corner point A: (20, 40) -> Cost = 20
Corner point B: (40, 25) -> Cost = 25
Corner point C: (60, 0) -> Cost = 30
Therefore, the minimum cost is $20, which occurs when John buys 20 boxes of Cereal A and 40 boxes of Cereal B.
The lines below are parallel if the slope of the solid line is 2, what is the slope of the gas line?
Answers
Answer:
2
Step-by-step explanation:
Parallel lines have the same slope. Hope this helps!
Plzzzzzzz help right answer gets brainly
Answers
Answer:
C:"As time dragged on, she sang to pass the hours."
Step-by-step explanation:
Find the missing side length in the image below
Answers
The time is takes an object to move a certain distance is inversely propotional to the speed that it's travelling at.
it takes a car 12 minutes to travel the length of a road if it goes at an average speed of 60mph
a) How long would it take the car to travel down the same road if it travelled at an average speed of 30mph
B) if it takes the car 36 minutes to drive down the road, what will it's average speed
Answers
Answer:
a) 24 minutes
b) 20 mph
Step-by-step explanation:
Speed - distance -time:time= 12 min
\(\sf = \dfrac{12}{60} \ hour\)
speed = 60 mph
\(\sf \boxed{distance =speed * time}\)
\(\sf = 60 * \dfrac{12}{60}\\\\ = 12 \ m\)
a) Speed = 30 mph
\(\sf \boxed{time = \dfrac{distance}{speed}}\)
\(\sf = \dfrac{12}{30}\\\\ = \dfrac{2}{5} \ hour\\\\ = \dfrac{2}{5}*60\\\\= 2 * 12\\\\= 24 \ minutes\)
It will take 24 minutes.
c) time = 36 minutes
\(\sf = \dfrac{36}{60} \hour\)
\(\sf \boxed{speed= \dfrac{distance}{time}}\)
\(\sf = \dfrac{12}{ \dfrac{36}{60}}\\\\\\ = 12* \dfrac{60}{36}\\\\= 20 \ mph\)
Average speed = 20 mph
Roddy adjusts his simulation, making it easier and easier to win. Eventually, he finds that if the random numbers are between 1 and 4, and the contest only runs for 4 days, about 39 out of 10,000 people are likely to win (by getting 4,4,4,4).
He runs this simulation repeatedly to be sure and then sets aside 39 concert tickets, plus a few more to be safe.
When the contest is launched into the real world, though, what factors might affect the actual number of winners? Select all that apply.
The random chance of the number generator
People who get a 1 and then stop playing the game
The number of people who participate in the game
The results of his simulated trials
People who get a few 4s and then stop playing the game
Answers
Answer:
Option A, B, C and D should be your answer.
Step-by-step explanation:
I'm sorry if my answers are wrong.
PLEASE HELP ASAPPPP WILL GIVE BRAINLIEST 20+ POINTS
show ur work
Create a factorable polynomial with a GCF of 6. Rewrite that polynomial in two other equivalent forms. Explain how each form was created.
Answers
9514 1404 393
Answer:
6a +12b6(a +2b)3(2a +4b)Step-by-step explanation:
For the GCF to be 6, terms of the polynomial must be multiples of 6, but must have no other common factors. A simple polynomial that meets this requirements is ...
6a +12b
One equivalent form can be had by factoring 6 from the terms:
6(a +2b)
Since 6 has a factorization of its own, another equivalent form can be had by factoring out one of the factors of 6. The ones of interest are 2 and 3, so we can write ...
2(3a +6b)
or ...
3(2a +4b)
Answer:
6a+12b6(A+2b)3(2a+4b)Step-by-step explanation:
hope it helps
compare these decimals 0.7______0.6999
1. >
2.<
3.=
Answers
2. Paula learned to play a total of 10 pieces over the course of 5 weeks of piano lessons. In all, how
many weeks of piano lessons will Paula need before she will be able to play a total of 18 pieces?
Answers
Select the correct answer. Rewrite the following expression.
Answers
The solution of the expression, \(x^{\frac{9}{7} }\) is \(\sqrt[7]{x^{9} }\).
How to solve an expression?An expression is a combination of numbers, variables, functions such as addition, subtraction, multiplication or division etc.
Therefore, let's rewrite the expression to it's equivalent expression.
An equivalent expression is an expression that has the same value or worth as another expression, but does not look the same.
Therefore, using indices rule
\(x^{\frac{9}{7} }\) = \(\sqrt[7]{x^{9} }\)
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A bowl contains 7 marbles of various colors consisting of green, brown, and red.
Tim knows that in the bowl, 3 marbles are green and 1 marble is brown.
What is the probability that a randomly selected marble will be red?
Answers
Answer:
3/7
Step-by-step explanation:
7-4=3
I GIVEEEEE BRAINLILSTE
Answers
Answer:
Answer: 39
Step-by-step explanation:
Volume of a Pyramid
Given a pyramid of triangular base area Ab and height H.
The volume of the pyramid is calculated by the formula:
\(\displaystyle V=\frac{1}{3}A_b\cdot H\)
The perfume bottle is in the shape of a triangular pyramid with a height of H=9 inches. The area of the base is Ab=13 (square) inches.
Calculating the volume by substituting the given values:
\(\displaystyle V=\frac{1}{3}13\cdot 9\)
\(\displaystyle V=39 \ in^3\)
Answer: 39
A group of friends were working on a student film that had a budget of $640. They spent all their budget on equipment and props. They used 15% of their budget on props. How much money did they spend on equipment?
Answers
Step-by-step explanation:
Amount spend on props and equipment = 800
Percentage of amount spend on equipment = 83%
So,
Percentage of amount spend on props =
Therefore,
Amount spend on props
Have an amazing day :)
Will give brainlst! Pls help
Answers
Answer:
As the study time increases the test score increases
Step-by-step explanation:
Nonlinear because the points aren't going in a straight line, they points fluctuate over time
Make the following conversion.50 acres to square feet ?Ft^2
Answers
Recall that:
\(1\text{ acre= }43560\text{ square f}eet.\)Therefore:
\(50\text{ acres= 50}\times43560\text{ square f}eet=2178000\text{ square f}eet.\)Answer: 2178000 square feet (ft^2).
which of the following is equivalent to x^2 -5x +6
Answers
Hello!
x² - 5x + 6
= (x² - 2x) + (-3x + 6)
= x(x - 2) - 3(x - 2)
= (x - 2)(x - 3)
Magic Realm, Inc., has developed a new fantasy board game. The company sold 15,000 games last year at a selling price of $20 per game. Fixed costs associated with the game total $182,000 per year, and variable costs are $6 per game. Production of the game is entrusted to a printing contractor. Variable costs consist mostly of payments to this contractor.
Required:
1) Prepare a contribution format income statement for the game last year and compute the degree of operating leverage.
2) Management is confident that the company can sell 18,000 games next year (an increase of 3,000 games, or 20%, over last year).
Compute:
a) The expected percentage increase in net operating income for next year.
b) The expected total dollar net operating income for next year.
Answers
The expected total dollar net operating Income for next year = $70,000
1) The contribution format income statement for the game last year, and the degree of operating leverage is computed below:
Contribution format income statement for the game last year Sales (15,000 × $20) = $300,000
Variable expenses (15,000 × $6) = $90,000
Contribution margin = $210,000
Fixed expenses = $182,000Net operating income = $28,000
Degree of operating leverage = Contribution margin / Net operating income= $210,000 / $28,000= 7.5 2)
The expected percentage increase in net operating income for next year:
The expected sales in next year = 18,000
games selling price per game = $20
Therefore, Total sales revenue = 18,000 × $20 = $360,000
Variable expenses = 18,000 × $6 = $108,000
Fixed expenses = $182,000
Expected net operating income = Total sales revenue – Variable expenses – Fixed expenses
= $360,000 – $108,000 – $182,000= $70,000
The expected percentage increase in net operating income = (Expected net operating income - Last year's net operating income) / Last year's net operating income*100= ($70,000 - $28,000) / $28,000*100= 150%
The expected total dollar net operating income for next year = $70,000
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What is the slope of the line that contains the points (8, 4) and (−4, 8)?
3
−3
one third
negative one third
Answers
Answer: negative one third
Answer:
-1/3
Explanation:
There's a formula that I'll use to find the slope - it's called the slope formula.
\(\sf{m=\dfrac{y_2-y_1}{x_2-x_1}}\)
Plug in the data:
\(\sf{m=\dfrac{8-4}{-4-8}}\\\\\sf{m=\dfrac{4}{-12}\)
Simplify:
\(\sf{m=-\dfrac{4}{12}}\)
\(\sf{m=-\dfrac{1}{3}}\)
Hence, the slope is negative one-third.
A shipping box is 36 inches by 24 inches by 18inches; how many cubic feet can it hold?
1. 8
2. 9
3. 10.
4. 12
5. 15
Answers
To find the cubic feet of an object, multiply the length, width and height and divide the resulting number by 1,728.
Within this situation,
36x24x18= 15,552
Next divide the product,
15,552/ 1,728= 9
The volume of the given shipping box is 9 feet square thus it can hold 9 feet square so option (2) will be correct.
What is volume?Volume is the scalar quantity of any object that specified occupied space in 3D.
For example, the space in our room is referred to as volume.
Volume has units of cube example meter³,cm³, etc.
Given the dimension of the box,
36 inches by 24 inches by 18 inches
Since,1 feet = 12 inches so,
36 inches = 3 feet,24 inches = 2 feet and 18 inches = 1.5 feet
Volume = 3 × 2 × 1.5 = 9 feet²
Hence "The volume of the given shipping box is 9 feet square thus it can hold 9 feet square".
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What is definition of and INDEPENDENT Vairable?
Answers
Answer:
a variable (often denoted by x) whose variation does not depend on that of another.
Step-by-step explanation:
Answer:
An independent variable is exactly what it sounds like. It is a variable that stands alone and isn't changed by the other variables you are trying to measure. For example, someone's age might be an independent variable.
58 children go into the hall for a concert. There are six seats in a row. How many rows are needed to seat everyone?
Answers
To seat all 58 children in the hall for the concert, 10 rows are needed.
1. Determine the number of children per row: Since there are 6 seats in a row, each row can accommodate 6 children.
2. Divide the total number of children by the number of children per row: Divide 58 (the total number of children) by 6 (the number of children per row).
58 ÷ 6 = 9 with a remainder of 4
This means that 9 full rows can be formed, and there will be 4 children left after the last full row.
3. Check if there are any remaining children: Since there are 4 children remaining, they cannot form a complete row by themselves.
4. Calculate the number of rows needed: Add 1 to the number of full rows to account for the remaining children.
9 full rows + 1 row for the remaining children = 10 rows in total.
Therefore, to seat all 58 children in the hall for the concert, 10 rows are needed.
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Two integers have a difference of 8. a. What is the maximum product of these integers? b. What are the numbers that produce the maximum product?
Answers
a) There is no maximum product for two integers such that they have a difference of 8.
b) There are no values for two integers such that the product is a maximum.
How to determine the integers behind a given condition
In this question we must determine two integers such that exists a difference of 8 and the product is the maximum possible. Thus, we describe the situation by the following equations:
x - y = 8
p = x · y
x, y > 0, where p is also an integer.
First, substitute y in the second equation:
p = x · (x - 8)
p = x² - 8 · x
Second, graph the resulting expression (please see the graph).
According to the graph, the quadratic equation is monotonously crescent for x > 0, then there is no a maximum product as x and y diverges to the infinity.
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Choose the correct graph of the function
Answers
What is the solution to this inequality? 1/3x − 7>−4
Answers
Add 7 to both sides.
1/3x > −4 + 7Add −4 and 7 to get 3.
1/3x > 3Multiply both sides by 3, the reciprocal of 1/3 . Since 1/3 is >0, the direction of inequality remains the same.
x > 3 × 3Multiply 3 and 3 to get 9.
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Angle C has a measure of π/6 radians. A circle with a radius length of 10 inches is centered at Angle C's vertex. What is the length of the subtended arc along this circle?
 inches
Answers
Answer:
Step-by-step explanation:
arc length = rθ = 10·π/6 = 5π/3 ≅ 5.2 inches
pleaseeee helppp me nowwww:(((
Answers
Answer:
I can't see anything to read
Answer:
um
Step-by-step explanation:
explaino moreo
Shown below is a wooden six-sided (hexagon) frame. Which of
the following best approximates the slopes of the six line
segments?
O The slopes are approximately -1.4, 0, and 1.4.
O The slopes are approximately -1.7, 0, and 1.7.
O The slopes are approximately -1.8, 1, and 1.4.
O The slopes are approximately -1.7,0, and 1.4
Answers
The slopes are best approximated as: The slopes are approximately -1.7, 0, and 1.7.
How to find the slope between two coordinates?The formula to find the slope between two coordinates is expressed as:
Slope = (y₂ - y₁)/(x₂ - x₁)
The slope of each line above the x-axis are:
Slope 1 = (5 - 0)/(5 - 8)
Slope 1 = -1.7
Slope 2 = (5 - 5)/(5 - (-2))
Slope 2 = 0
Slope 3 = (5 - 0)/(-2 - (-5))
Slope 3 = 5/3 = 1.7
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TWO ANGLES OF A TRINGLE measures 12 degrees and 40 degress
Answers
Answer:
Ok, so what is the qns?
Step-by-step explanation: