Considering the given functions, we have that:
1) The inverse of function f(x) is defined by: [tex]f^{-1}(x) = \frac{4x^2}{9} + 1[/tex]
2) It is shown in the Item 2 section that the function is one-to-one.
3) The functions are not inverses, due to the result of their composition.
Item 1 - Inverse functionThe find the inverse of a function, we exchange x and y on the original function, and then isolate y.
The original function is given by:
[tex]y = 3\frac{\sqrt{x - 1}}{2}[/tex]
Exchanging x and y, we have that:
[tex]x = 3\frac{\sqrt{y - 1}}{2}[/tex]
Now we do the procedures to isolate y, as follows:
[tex]3\sqrt{y - 1} = 2x[/tex]
[tex]\sqrt{y - 1} = \frac{2x}{3}[/tex]
[tex](\sqrt{y-1})^2 = \left(\frac{2x}{3}\right)^2[/tex]
[tex]y - 1 = \frac{4x^2}{9}[/tex]
[tex]y = \frac{4x^2}{9} + 1[/tex]
[tex]f^{-1}(x) = \frac{4x^2}{9} + 1[/tex]
Item 2 - One-to-one functionTo verify if the function is one-to-one, we first find the numeric values at x = a and x = b, then:
f(a) = -2/(2a - 1).f(b) = -2/(2b - 1).Now we have to verify how many outcomes of a and b are possible for them to be equal. Since they have the same numerator, they will be equal when the denominators are equal, that is:
2a - 1 = 2b - 1.
Simplifying the one and then by 2:
a = b.
Only equal when a = b, hence the function is one-to-one. (this is the algebraic proof).
Item 3 - Are they inverses?The functions will be inverses if the composition of the functions result in x, that is, if inputting function g(x) into function f(x) results in x.
The result of the composition is given by:
f(g(x)) = f(x/3 + 5) = 3(x/3 + 5) - 5 = x + 15 - 5 = x + 10.
x + 10 is different of x, hence the two functions are not inverses.
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a random sample of 1000 people from california (population over 35,000,000) and 1000 people from rhode island (population just over 1,000,000) was taken by the gallup poll. which is correct?
Option D is the right choice.
The Gallup poll randomly selected 1000 individuals from Rhode Island (population just over 1,000,000) and 1000 individuals from California (population over 35,000,000). Because both samples have the same sample sizes, it can be assumed that they will be roughly equally accurate.
Taking a poll in the US. The department of Gallup that routinely conducts public opinion polls is called The Gallup Poll. In the form of data-driven news, Gallup Poll findings, analysis, and videos are released every day.
A simple random sample is a subset of people chosen at random from a larger group, all of whom were chosen with the same probability. It is a method of choosing a sample at random. Random sampling makes sure that the findings you get from your sample should be close to what you would have gotten if you measured the complete population.
The Gallup survey randomly selected 1000 individuals from Rhode Island (population just over 1,000,000) and 1000 individuals from California (population over 35,000,000). Because both samples have the same sample sizes, it may be assumed that they will be roughly equally accurate.
Therefore, Option D is correct.
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COMPLETE QUESTION:
A random sample of 1000 people from California (population over 35,000,000) and 1000 people from Rhode Island (population just over 1,000,000) was taken by the Gallup Poll. Which is correct?
a) The sample from California will be more biased because the sampling fraction is smaller
b) The sample from Rhode Island will be substantially more accurate than that from California because a higher sampling fraction was chosen.
c) The sample from California will have a larger sampling error than from Rhode Island because less of the population is sampled.
d) Both samples will be approximately equally accurate because the sample sizes are the same
If 30 % of x =20 ,find value of x
Answer:
[tex]\sf 66\dfrac{2}{3}[/tex]
Step-by-step explanation:
Finding value of 'x':
[tex]\sf 30 \% \ of \ x = 20\\\\\\ \dfrac{30}{100}*x = 20\\\\[/tex]
To find the value of x, multiply both sides with the multiplicative inverse (Reciprocal) of 30/100.
[tex]\sf x = 20*\dfrac{100}{30}\\\\x = \dfrac{200}{3}\\\\x= 66\dfrac{2}{3}[/tex]
Trey rented a truck for one day. there was a base fee of $15.95, and there was an additional charge of 99 cents for each mile driven. trey had to pay $188.21 when he returned the truck. for how many miles did he drive the truck?
I need help I’ll give Brainliest
The perimeter of the triangle DEF is half the perimeter of triangle ABC.
We are given a triangle ABC with mid-points D,E, and F of AB, BC and CA respectively.
DEF forms a triangle.
We have to find the relationship between the perimeter of triangle ABC and triangle DEF.
Let us consider sides AB,BC, and CA as 2x, 2y and 2z.
We know that triangle ABC is divided into four equal triangles of sides which are half of the sides of triangle ABC.
Hence,
DE = x,
EF = y, and
DF = z
Hence,
Perimeter of triangle ABC = 2x + 2y + 2z = 2(x + y + z)
Perimeter of triangle DEF = x + y + z
Hence, we can say that the perimeter of triangle DEF is half the perimeter of triangle ABC.
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5^5=5x5x5x5x5=
i put 3125 but it wont work can someone help me with this
Answer: 4 i think
Step-by-step explanation:
Suppose that W and X are points on the number line.
If WX=12 and W lies at 7, where could X be located?
If there is more than one location, separate them with commas.
Suppose that W and X are points on the number line. The X can be located as 21 or -5.
What is a line?A line, according to Euclid, is a segment connecting two points that may be stretched forever in any direction.'
There is currently a lack of information to determine whether or not points W or X came first in time.
Consider two examples
The first scenario: W predates X.
=7-12
=-5
Example 2: If X comes before X, then X is at
=|7+12|
=19
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15. Explain the steps in finding the midpoint of the points (-1,2) and (3,-6). Then state what the midpoint is.
The given points are (-1,2) and (3,-6).
We use the midpoint formula
[tex]M=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})[/tex]Where,
[tex]\begin{gathered} x_1=-1 \\ x_2=3 \\ y_1=2 \\ y_2=-6 \end{gathered}[/tex]Replacing all these values, we have
[tex]\begin{gathered} M=(\frac{-1+3}{2},\frac{2-6}{2}) \\ M=(\frac{2}{2},\frac{-4}{2}) \\ M=(1,-2) \end{gathered}[/tex]Therefore, the midpoint is (1,-2).While measuring a cloth of length 1.5 meters, it wasmeasured as 1.53 meters. Find the percentage error inmeasuring the cloth.
we have that
1.5 meters represents 100%
so
Applying proportion
Find out how much percentage represents the difference (1.53-1.50=0.03 m)
100/1.5=x/0.03
solve for x
x=(100/1.5)*0.03
x=2%
therefore
the answer is 2%There is a line that includes the point (6,1) and has a slope of 1/4 . What is its equation in point-slope form?
Answer:
y - 1= 0.25(x - 6)
Step-by-step explanation:
because.
A regulation-sized gymnasium is 80 feet wide and 100 feet long. If a scale model of the gymnasium is 12 inches long, how wide is the model?
The width of the scale model is 9.6 inches long
How to determine the width of the model?The dimensions of the regulation-sized gymnasium is given as
80 feet wide by 100 feet long
This means that
Width = 80 feet
Length = 100 feet
For the scale model, we have
Length = 12 inches
The scale can be represented as
Scale = Length : Width
This means that
12 inches : Width = 100 feet : 80 feet
Express as fraction
So, we have
Width/12 inches = 80 feet/100 feet
This gives
Width/12 inches = 0.80
Multiply both sides by 12 inches
Width = 9.6 inches
Hence, the width is 9.6 inches
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15. Georgia works at the circus selling cotton
candy. The table shows the linear relationship
between the number of
cotton candy bags remaining Hours
in her stand as a function of
the hours she has been
selling. Write a linear
equation in slope-intercept/
form to model the situation.
3
5
12
16
Cotton
Candy
90
78
36
12
The linear equation in slope-intercept/ form to model the situation is 6x + y - 108 = 0
Given:
Georgia works at the circus selling cotton candy. The table shows the linear relationship between the number of cotton candy bags remaining Hours in her stand as a function of the hours she has been selling.
Table :
cotton candy
3 90
5 78
12 36
16 12
Let x,y be cotton,candy.
(x1,y1) = (3,90)
(x2,y2) = (5,78)
slope m = y2-y1/x2-x1
= 78 - 90/5 - 3
= -12/2
m= -6
Slope intercept form y=mx+c
y = -6x + c
substitue (3,90) in y = -6x + c
90 = -6(3) + c
c = 108
substitute c value in y = -6x + c
y = -6x + 108
6x + y - 108 = 0
Therefore the linear equation in slope-intercept/ form to model the situation is 6x + y - 108 = 0.
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I tried everything but couldn't figure out what x equals.
Answer:
x = 17Step-by-step explanation:
The two marked angles are same side interior angles.
As per definition, these angles are supplementary, in other words they sum to 180°.
Show this as equation and solve for x:
6x + 8 + 4x + 2 = 18010x + 10 = 18010x = 170x = 17Answer:
✨ sorry i need points, bro-♡
Step-by-step explanation:
Male and female high school students reported how many hours they worked each week in summer jobs. The data is represented in the following box plots:
Identify any values of data that might affect the statistical measures of spread and
center
A. The zero hour mark on both plots prevents the graphs from being balanced.
B. The median is near the center of the IQR for both males and females.
C. There is not enough evidence to see any effects on spread or center.
D. The males have a suspected significant high outlier.
The values of data that might affect the statistical measures of spread and center is that: D. The males have a suspected significant high outlier.
What is a box and whisker plot?A boxplot is also referred to as box and whisker plot and it can be defined as a type of chart that can be used to graphically represent the five-number summary of a data set with respect to locality, skewness, and spread.
In Mathematics, the five-number summary of any box and whisker plot include the following:
MinimumFirst quartileMedianThird quartileMaximumBy critically observing the box and whisker plots which represent the number of hours the students worked in summer jobs, we can reasonably infer and logically deduce that male students have a significant high outlier because the maximum value (37) is very far from the third quartile (15).
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The speedometer in Erin’s car is accurate within 2 miles per hour. It currently reads
56 mph. Write and solve an absolute value inequality to represent the situation. At what speeds could Erin be driving?
Answers:
The absolute value inequality is [tex]|\text{x}-56| \le 2[/tex] which solves to [tex]54 \le \text{x} \le 58\\\\[/tex] Her true speed could be anything between 54 mph and 58 mph, including both endpoints.===================================================
Explanation:
x = Erin's true speed
Her true speed x is within 2 mph of 56 mph.
This means the interval for x is [tex]56-2 \le \text{x} \le 56+2[/tex] which simplifies to [tex]54 \le \text{x} \le 58[/tex]
She could be going as slow as 54 mph, or as fast as 58 mph, or something in between.
The expression |x-56| measures the distance from x to 56 on the number line. Recall absolute value is used to ensure the distance isn't negative.
We want this distance to be within 2 units, i.e. we want the distance to be 2 or smaller.
[tex]\text{distance on number line} \le 2\\\\|\text{x}-56| \le 2[/tex]
----------------
Here's what the steps look like to solve that absolute value inequality
[tex]|\text{x}-56| \le 2\\\\-2 \le \text{x}-56 \le 2\\\\-2+56 \le \text{x}-56+56 \le 2+56\\\\54 \le \text{x} \le 58\\\\[/tex]
For the second step, we use the rule that |x| < k is the same as -k < x < k where k is some positive number.
In the third step, I added 56 to all three sides to isolate x fully.
Seven minus a number divided by eleven is greater than or equal to six.
Answer:
7 - x/11 ≥ 6
x ≤ 11
Step-by-step explanation:
7 - x/11 ≥ 6
(77- x)/11 ≥ 6
(77- x) ≥ 66
- x ≥ -11
11 ≥ x
x ≤ 11
0, 6) and (3, 0). What is the slope?
Answer:
m=-2
Step-by-step explanation:
y=mx+b you have to use (y2-y1)/(x2-x1)
Please help me …………
Answer:
[tex] \frac{5}{8} = \frac{22}{x - 6} \\ x≠6 \\ 5(x - 6) = 176 \\ 5x -30 = 176[/tex]
Give answer to 1 decimal place
The side CD of the right angle triangle is 8.7 cm.
How to find the side of a right angle triangle?A right angle triangle is a triangle that has one of its angles as 90 degrees.
The side CD can be found using trigonometric ratios to find length CB and DB, then subtract the length DB from CB to get the required length CD(x).
An angle bisector bisect ang angle into two equal halves.
Therefore, AD is an angle bisector that bisects the angle CAB.
Hence,
∠CAD = 19°
∠DAB = 19°
using trigonometric ratios,
tan 19 = opposite / adjacent
tan 19 = DB / 20
DB = 20 tan 19
DB = 6.88655226579
DB = 6.887 cm
Therefore,
tan 38° = CB / 20
CB = 20 tan 38
CB = 15.6257125301
Therefore,
CD = CB - DB
CD = 15.626 - 6.887
Therefore,
CD = 8.739
CD = 8.7 cm
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Rewrite the following equation of a line to express it in slope-intercept form:
4x - 2y = 14
A:2y=14-4x
B:x=5y+3.5
C:y=2x-7
D:y=-2x-7
Answer:
C
Step-by-step explanation:
the equation of a line in slope- intercept form is
y = mx + c ( m is the slope and c the y- intercept )
given
4x - 2y = 14 ( subtract 4x from both sides )
- 2y = - 4x + 14 ( divide through by - 2 )
y = 2x - 7 ← in slope- intercept form
The school decides to put a chain link fence around a soccer field
area. The dimensions are 13 yards by 13 yards. If the contractor
charges $6 per yard, what is the total fence cost?
Reason:
L = length = 13
W = width = 13
The rectangle has a perimeter of 2*(L+W) = 2*(13+13) = 52 yards. This is the amount of fencing needed. Since the contractor charges $6 per yard of fencing, the total cost is 52*6 = 312 dollars
Can somebody please explain what domain and range is on a graph and how to find them.
Answer:
Domain: x-values
Range: y-values
-
In the graph,
Domain: {-6, 4} or you could write {-6,-5,-4,-3,-2,-1,0,1,2,3,4}
Range: {-9,7} or you could write {-9,-8,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7}
-
(Please look at the picture attached, and read what's written above. You'll get a better understanding)
Two sisters like to compete on their bike rides. Ciara can go 4 mph faster than her sister, Colleen. If it takes Colleen one hour longer than Ciara to go 80 miles, how fast can Colleen ride her bike?
Question:
Two sisters like to compete on their bike rides. Ciara can go 4 mph faster than her sister, Colleen. If it takes Colleen one hour longer than Ciara to go 80 miles, how fast can Colleen ride her bike?
Explanation:
Note that Ciara can go 4mph faster than Collen.
Let's take the speed to ride a bike for Collen to be x mph.
The speed to ride a bike for Ciara will be (x + 4) mph
Now, to cover a distance of 80 miles, Collen takes 1 hour longer than Ciara. Applying the kinematics equation for motion at constant speed:
[tex]v=\frac{d}{t}[/tex]where v is velocity, d is distance and t is time, we can solve this equation for the time t:
[tex]t=\frac{d}{v}[/tex]Here time Collen takes to cover a distance of 80 miles in 1 hour is more than that taken by Ciara, hence:
Time taken by Collen:
[tex]t=\frac{80}{x}[/tex]Time taken by Ciara:
[tex]t=\frac{80}{x+4}[/tex]The equation for the difference in time:
[tex]\frac{80}{x}\text{ - }\frac{80}{x+4\text{ }}=1[/tex]Solve the equation for the difference in time to get the value of x which is Collen speed:
[tex]x^2+4x\text{ - 320=0}[/tex]Solving the quadratic equation by the quadratic formula where a=1,b=4, and c=-320, we get:
[tex]x=16[/tex]Answer: Colleen's speed is 16 mph.
The perimeter of the rectangle is
A) 9.49
B) The p
C) 18.97
The area of the rectangle
A) 18.5
B) 20
C) 40
Please answer ASAP
The area of the rectangle = 13.12 square units.
Perimeter of the rectangle = 21.77 units.
What exactly is a rectangle?A rectangle is a quadrilateral with parallel sides that are equal to one another and four 90-degree vertices. degrees.As a result, it is sometimes referred to as an equiangular quadrilateral.A rectangle is also known as a parallelogram because its opposite sides are both equal and parallel.In response to the given question;
To determine the perimeter of the rectangle ABCD, first determine the length of each side.
Determine the lengths DA and DC using the distance formula.
(x2 - x1)2 + (y2 - y1)2 = DA
Enter the coordinates of points D and A.
DA = √(8 - 1)² + (2 - 0)²
= √49 + 4
= √53 units.
Now, we have to find length of DC
Here , DC = √(3 - 1)² + (3 - 0)²
= √4 + 9
= √13 units.
Here,
DA = CB = √53 units.
And,
DC = AB = √13 units.
So Perimeter of rectangle = DA + CB + DC + AB
= 2√53 + 2√13 units
= 2(√53 + √13) units.
= 21.77 units.
To find area,
Area = (1/2) base × height
Here area,
= (1/2) CB × CD
= (1/2) × √53 × √13
= 26.24/2
∴ Area = 13.12 sq. unit.
Thus, area = 13.12squnits
perimeter = 21.77 units .
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What is the equation of the line that passes through the point (-2, -4) and
has a slope of -1/2
2 x 10 to the power of 5 / 4 x 10 to the power of 2 in standard form
The result of 2 x 10 to the power of 5 ÷ 4 x 10 to the power of 2 in the standard form is 5×[tex]10^{2}[/tex]
The first number in standard form = 2×[tex]10^{5}[/tex]
The second number in standard form = 4×[tex]10^{2}[/tex]
Scientific notation of the number is a method to represent a very large number or very small number in a simple way. In scientific notation a very whole number from 1 to 10 is multiplied by the power of 10
The first number divided by the second number
2×[tex]10^{5}[/tex] ÷ 4×[tex]10^{2}[/tex] = (2/4)×[tex]10^{5-2}[/tex]
= 0.5×[tex]10^{3}[/tex]
= 5×[tex]10^{2}[/tex]
Hence, the result of 2 x 10 to the power of 5 ÷ 4 x 10 to the power of 2 in the standard form is 5×[tex]10^{2}[/tex]
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maria and cindy both arrived early for work. cindy arrived 23 1/2 minutes earlier than maria. if the total number of minutes that they were early was 38, find how many minutes each girl arrived ahead of time
If Maria and Cindy both arrived early for work, Cindy arriving 23(1/2) minutes earlier than maria, and the total number of minutes that they were early by, was 38 minutes, then Maria arrived ahead of stipulated time by 14.5 minutes, while Cindy arrived earlier by 38 minutes, determined by solving a linear equation.
As per the question statement, Maria and Cindy both arrived early for work, Cindy arriving 23(1/2) minutes earlier than maria, and the total number of minutes that they were early by, was 38 minutes.
We are required to calculate the individual minutes by which, each of Maria and Cindy arrived earlier by.
To solve this question, let us assume that Maria arrived early by "x" minutes, and already given that Cindy arrived 23(1/2) minutes earlier than maria.
Since the total number of minutes that they were early by, was 38 minutes, we can form a linear equation, based on our assumption and the condition provided in the question statement, which goes as,
[{x + 23(1/2)} = 38]
Or, [(x + 23.5) = 38]
Or, [(x +23.5) = 38]
Or, [x = (38 - 23.5)]
Or, [x = 14.5]
That is, Maria arrived ahead of stipulated time by 14.5 minutes.
Linear Equation: A linear equation is a mathematical statement expressing the equality between two expressions containing variable(s), and it gives a straight line when plotted on a graph.To learn more about Linear Equations, click on the link below.
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The height of men is a normally distrubuted variable with a mean of 68 inches and a standard deviation of 3 inches.**Round answers to ONE decimal place**a.) What is the minimum height you could be to be considered in the top 10% of tallest men? b.) What is the tallest you could be to be considered in the shortest 15% of men?
For this problem, we are given the mean and standard deviation for the height of men. We need to calculate the minimum height to be considered in the top 10% of tallest men, and the maximum height to be considered in the shortest 15% of men.
The first step we need to solve this problem is to calculate the z-score. The z-score can be found by using the following expression:
[tex]z=\frac{x-\mu}{\sigma}[/tex]For the first situation, we want a z-score for the top 10% of tallest men. This means that we need to go on the z-table and find the z-score that represents 90% of probability to the left because this will give us the minimum height to be at the 10% tallest. From the z-table we have:
[tex]z=1.29[/tex]Now we can use the z-score expression to find the value of x. We have:
[tex]\begin{gathered} 1.29=\frac{x-68}{3} \\ x-68=3.87 \\ x=3.87+68 \\ x=71.87 \end{gathered}[/tex]The man should be at least 71.85 inches tall to be considered among the 10% of tallest men.
For the second situation, we have something similar. We need to find the maximum height for a man to be considered between the 15% of men. We need to go into the z-table and find the z-score that produces a result close to 0.15. We have:
[tex]z=-1.04[/tex]Now we need to use the z-score expression to determine the height:
[tex]\begin{gathered} -1.04=\frac{x-68}{3} \\ x-68=-3.12 \\ x=68-3.12 \\ x=64.88 \end{gathered}[/tex]In order to be considered among the smallest men, someone needs to be 64.88 inches tall.
plse answer the questions
Answer:
a. [tex]3^4[/tex]
b. [tex]7^5[/tex]
c. [tex]10^2[/tex]
d. [tex]5^7[/tex]
Step-by-step explanation:
When multiplying or dividing numbers with exponents, the rule is that you add the exponents together if multiplying, or subtract them if dividing. This is only possible with numbers of the same base (the number before the exponent).
[tex]3^2[/tex] multiplied by [tex]3^2[/tex] is [tex]3^4[/tex] because the exponents, 2 and 2, are added together. The bases remain unchanged.
[tex]7[/tex] times [tex]7^4[/tex] is [tex]7^5[/tex] because another way to write [tex]7[/tex] is [tex]7^1[/tex], and the exponents can be added together from there.
The same logic can be applied to the division questions, but with subtraction instead of addition.
Owen has 24 pens and 18 notebooks. He wants to divide the pens and notebooks into equal groups. What is the greatest number of groups he can make using all the pens and notebooks?
Answer:
6
Step-by-step explanation:
You have to think of the GCF (Greatest Common Factor)
List the numbers that multiply 24: 1,2,3,4,6,8,12,24
List the number that multiply 18: 1,2,3,6,9,18
You look at the greatest number that they both have in common.
Thabo is conducting an experiment in which the temperature is measured
carefully. The temperature was 7°C at the end of the first minute, and then it
increased by 6°C every minute after that.
4.2.1 What will the temperature be after the third minute?
4.2.1
Determine a formula to calculate the temperature (7) after
m minutes.
Answer:
after 3 min the temp will be 19°
formula is (6•m)+1
