Answers
The value of f(g(x))=\(\frac{7x}{1-x}\) and g(f(x))=\(6(\frac{7x}{6-7x})\). The function f(g(x)) = g(f(x)) are inverse of each other.
What is the value of a function called?In mathematics, an argument of a function is a value provided to obtain the function's result. It is also called an independent variable.
1. To find the f(g(x)), we need to substitute g(x) into the expression for f(x):
\(f(g(x))= \frac{(f(x))}{(6+g(x))} \\f(g(x)) =\frac{(6x/(1-x))}{6+(6x)/(1-x)} \\f(g(x))=\frac{(7x)}{(1-x)}\)
To find g(f(x)), we need to subsititute f(x) into the expression for g(x):
\(g(f(x))= \frac{6(f(x))}{(1-f(x))} \\g(f(x)) =\frac{6((x/6)+x)}{(1-((x/6)+6)} \\g(f(x)) =\frac{6(7x/6)}{(1-(7x/6))} \\g(f(x))=\frac{6(7x)}{(6-7x)}\)
2. The fact that f(g(x)) = g(f(x)) tells us that these two functions are inverses of each other. This means that if we apply f and then g to a number, we get back the same number, and if we apply g and then f to a number, we also get back the same number.
In other words, these two functions "undo" each other's operations.
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a. f(g(x)) = x/(1 - x) and g(f(x)) = x/(1 - x/6)
b. f(x) and g(x) are not inverses of one another.
What is function?
In mathematics, a function is a relationship between two sets of elements, called the domain and the range, such that each element in the domain is associated with a unique element in the range.
(a)
To find f(g(x)), we substitute g(x) into f(x) in place of x:
f(g(x)) = g(x)/6 + g(x) = [(6x)/(1 - x)]/6 + (6x)/(1 - x) = x/(1 - x)
To find g(f(x)), we substitute f(x) into g(x) in place of x:
g(f(x)) = (6f(x))/(1 - f(x)) = (6[x/6 + x])/(1 - [x/6 + x]) = x/(1 - x/6)
(b)
Comparing f(g(x)) = x/(1 - x) and g(f(x)) = x/(1 - x/6), we can see that they are not the same expression. Therefore, f(x) and g(x) are not inverses of one another. In fact, we can verify that:
f(g(x)) ≠ x
g(f(x)) ≠ x
Thus, there is no inverse relationship between the two functions.
When f(g(x)) = g(f(x)) = x, it means that the functions f(x) and g(x) are inverse functions of each other. This implies that composing one function with its inverse function results in the identity function. However, this is not the case for the functions f(x) and g(x) given in this question.
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Related Questions
Researchers at Manchester Metropolitan University in England determined experimentally that if a piece of toast is dropped from a 2.5 foot high table, the probability that it lands butter side down is 0.81.A) Explain what this probability means.
B) If you dropped 100 pieces of toast, will exactly 81 of them land butter side down? Explain
C) Maria decides to test this probability and drops 10 pieces of toast from a 2.5 foot table. Only 4 of them land butter side down. Should she be surprised? Describe how you would carry out a simulation to answer this question, assuming the probability that the toast lands butter side down is 0.81. Do not perform the simulation.
D) The dot plot displays the results of 50 simulated trials of dropping 10 pieces of toast. Does the simulation suggest that Maria should be surprised? Explain.
Answers
A) The probability that a piece of toast lands butter side down when dropped from a 2.5-foot high table is 0.81 means that out of all possible outcomes of dropping a piece of toast from that height, 81% of the time it will land with the butter side facing down.
B) No, it does not mean that exactly 81 out of 100 pieces of toast will land butter side down. The probability of a single piece of toast landing butter-side down is 0.81, so when 100 pieces of toast are dropped, we can expect around 81 of them to land butter-side down on average, but the actual number may vary due to randomness.
C) Maria may or may not be surprised, depending on her expectations and the level of significance she sets for her test. To carry out a simulation, we can use a random number generator to simulate dropping 10 pieces of toast with a probability of 0.81 of landing butter side down. We can repeat this simulation multiple times, say 1000 times, and count the number of times the simulation yields 4 or fewer pieces of toast landing butter side down. If this rarely happens (e.g., less than 5% of the time), we can say that Maria should be surprised.
D) The dot plot displaying the results of 50 simulated trials of dropping 10 pieces of toast can provide some information about the variability of the outcomes. If most of the dots are clustered around 8 or 9 (which would correspond to 80% or 90% of the pieces of toast landing butter side down), then Maria's result of only 4 out of 10 pieces landing butter side down would be considered unusual, and she should be surprised. However, if the dots are spread over a wide range, it would suggest that getting 4 out of 10 is not that uncommon, and Maria should not be surprised.
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Blue Street passes through the points (-2,6) and (3,-4) find the slope using the slope formula and then find the equation of the line using any method. Show your work.
Answers
Answer:
Christianity
May god Bless you
Step-by-step explanation:
A number between 10 and 100 is four times the sum of its digits. If the product of two digits is 8, find the sum of digits of the number.
Answers
Answer:
24 is the number. The sum of its digits is 6.
Step-by-step explanation:
There are four two-digit numbers the product of whose digits is 8: 18, 24, 42, and 81. Of these numbers, 24 is four times the sum of its digits:
24 = 4 × 6 = 4 × (2 + 4)
Find the slope and y-intercept of this linear function:
2x + x = 4(y - 1)
Answers
Answer:
slope: 3/4y-intercept: 1Step-by-step explanation:
Solve for y to put the equation in slope-intercept form.
3x = 4y -4 . . . . . eliminate parentheses, collect terms
3x +4 = 4y . . . . . add 4
y = 3/4x +1 . . . . . divide by 4
The slope is the x-coefficient: 3/4.
The y-intercept is the constant: 1.
can someone please tell me the answer thank you
Answers
Step-by-step explanation:
In the problem 5 is raised to an odd exponent(9). Thus, there is no way to halve it apart from doing 5^4.5, which equals 1397.54249, which is not a whole number.
Hope it helps <3
Rewrite the following in dx dz dy order.
Answers
\(x\) is minimized at \(x=y^2\), so the upper bound for \(z\) can be rewritten as
\(z = 2 - \dfrac x2 = 2 - \dfrac{y^2}2\)
so that in the integral with \(dz\), we have the range \(0 \le z \le 2 - \frac{y^2}2\).
Rewrite the plane equation as a function \(x=x(z)\).
\(z = 2 - \dfrac x2 \implies 2z = 4 - x \implies x = 4 - 2z\)
So the equivalent integral is the one in choice A,
\(\displaystyle \int_{-2}^2 \int_{y^2}^4 \int_0^{2-x/2} dz\,dx\,dy = \boxed{\int_{-2}^2 \int_0^{2-y^2/2} \int_{y^2}^{4-2z} dx\,dz\,dy}\)
What is the slope of the line represented by the given table?
X -14 0 7 14 21
y -6 2 6 10 14
Answers
if tantheta -cottheta =a and costheta +sintheta =b then 4b^2(2-b^2)
Answers
Answer:
= 4(1+sin2theta)²
Step-by-step explanation:
Given the equations
tantheta -cottheta =a... 1
costheta +sintheta =b .... 2
We are to find 4b^2(2-b^2)
Squaring both sides of equation 2
(costheta +sintheta)^2 = b^2
cos^2theta + 2costheta sintheta + sin^2 theta = b^2
(cos^2theta+ sin^2 theta) + 2costheta sintheta = b^2
1 + 2costheta sintheta = b^2
1 + sin2theta = b^2
substitute
4b^2(2-b^2)
= 4(1+sin2theta)(2-1+sin2theta)
4(1+sin2theta)(1+sin2theta)
= 4(1+sin2theta)²
Please help meeeeeee !!??? I begggggggggggggggggg it due tomorrow
Answers
Answer:
\( \frac{14}{16} \: and \: \frac{13}{16} \)
Step-by-step explanation:
So we have these two fractions
\( \frac{7}{8} \: and \: \frac{13}{16} \)
Does 8 go into 16? Yes. It goes in twice because 8 times 2 is 16. That means 16 is our common denominator.
Because of that, since we multiplied 8 by 2, we will do the same to the 7. 7 times 2 is 14.
Because of that we have 14/16 and 13/16 as our answers.
StartFraction 32 Over 8 EndFraction = StartFraction 28 Over x EndFraction
a.
x = 4
c.
x = 8
b.
x = 28
d.
x = 7
Answers
Answer:
d. x = 7
Step-by-step explanation:
\(\frac{32}{8} = \frac{28}{x}\)
Cross multiply, the denominator "8" is multiplied by numerator "28", while the numerator "32" is multiplied by denominator "x":
8 * 28 = 224
32 * x = 32x
\(32x = 224\)
Divide both sides by 32:
\(\frac{32}{32} = \frac{224}{32}\)
[x = 7]
8. The first three terms of a geometric sequence are ( x-6), 3x, and y. If the common ratio is 6, then the value of y is.
Answers
Answer:
The value of y is 216
(and the value of x is 12)
Step-by-step explanation:
The general formula for a geometric sequence is,
\(a_n = a_1(r)^{n-1}\)
Where n represents the nth term, a_1 is the first term and r is the common ratio,
we see that,
r = 6,
the first term is,
a_1 = (x-6)
the 2nd term is,
a_2 = 3x,
the 3rd term is,
a_3 = y, finding y,
first we find x, using the above given formula we have,
\(a_2 = a_1(6)^{2-1}\\3x = (x-6)(6^1)\\3x = 6x -36\\36 = 6x - 3x\\36 = 3x\\x=36/3\\x=12\)
x = 12,
Now, for y we can use the relation between a_3 and a_2,
\(a_3 = a_1(6)^{3-1}\\y = (x-6)(6)^2\\y = (12-6)(6^2)\\y = 6(6^2)\\y = 6^3\\y = 216\)
y = 216
Complete the table with the measure of angles c and d
Answers
Answer:
m < c = 98 degrees
m < d = 82 degrees
Explanation:
sum of supplementary angles are 180 degrees
mAccording to the figure,
98 + mm m
m < c + m < d = 180
m < d = 82
m < c = 180 - 82
m < c = 98 degrees
Therefore, m < c = 98 degrees and m < d = 92 degrees
what led to the development of the aviation industry
a. the development of texas’ wwII aircraft training facilities
b. the placement of the johnson space center
c. the petrochemicals industry
d. texas’ location and climate
Answers
Texas' location and climate led to the development of the aviation industry.
What is the significance of Texas in the development of the aviation industry?
The development of the aviation industry in Texas can be attributed to its favorable geographical location and climate. Texas is located in the southern part of the United States, making it a strategic location for air travel between the east and west coasts of the country. Additionally, its warm and dry climate provided ideal conditions for flight testing and training.
Reason what led to the development of the aviation industry :
Texas has a long history of aviation, with the Wright brothers conducting one of their earliest flight demonstrations in the state in 1910. The establishment of military bases during World War II and the subsequent development of aircraft training facilities in Texas further fueled the growth of the aviation industry in the state. Additionally, the placement of the Johnson Space Center in Houston helped to establish Texas as a hub for aerospace research and development. However, it is Texas' favorable location and climate that truly made it a prime location for the aviation industry. Its central location made it a strategic location for air travel and logistics, while its warm and dry climate provided ideal conditions for flight testing and training. Today, Texas remains one of the top states for aviation and aerospace industries, with major airports, aircraft manufacturers, and research institutions located throughout the state.
Therefore, option (d) is correct.
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I’m gonna die please
Answers
Answer:
The answer is B.
hope this helps
have a good day :)
Step-by-step explanation:
v=pi*r^2h
Answer:
5626.9m³
Step-by-step explanation:
V = πr²h
radius = 8
height = 28
to make it easier first step is to get the multiply the radius and the height but first multiple the radius by itself
=> 8 x 8 = 64
=> 64 x 28 = 1792
then multiply 1792 by π (3.14)
=> 1792 x 3.14 = 5626.88
when you round the answer it will be 5626.9
hope this helps and is right :)
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.
Let (-3, 4) be a point on the terminal side of 0. Find the exact values of sin 0, csc 0, and cot 0.
Answers
Therefore, sin θ = 4/5, csc θ = 5/4, and cot θ = -3/4 using trigonometric ratio.
Trigonometric ratio calculation.
We know that the point (-3, 4) is on the terminal side of the angle θ, and we can use the Pythagorean theorem to find the value of the hypotenuse:
r² = x² + y²
r² = (-3)² + 4²
r² = 9 + 16
r² = 25
r = 5
Now we can find the values of sin θ, csc θ, and cot θ:
sin θ = y/r = 4/5
csc θ = r/y = 5/4
cot θ = x/y = -3/4
Therefore, sin θ = 4/5, csc θ = 5/4, and cot θ = -3/4.
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evaluate f(x)=4|x|+3 for x=-5
Answers
f(-5) = 4 (answer)
Consider the figure below with circles R, C, and D below, where circle R and circle D are
congruent. If RA = 42 and CE = 32, then answer the questions below.
R
D
424
E
32
с
Part A: Determine CA + ED.
Part B: Determine the combined circumference of all three circles.
Answers
Answer:
a) CA + ED = 74
Step-by-step explanation:
The complete question is attached in the image
Solution
Here the two circles R and D are congruent and thus their radius will be equal
Hence, RA (radius of circle R ) = DE (radius of circle D)
Hence, CA + ED = CA + 42
Also, RC = CD
CA = CE + RA = 32
So, CA + EO = 32 +42 = 74
VE
If Joel will need $5,000 to buy a car in two years, then how much will he need to deposit now
(rounded to the nearest dollar) if his savings account pays an annual 10% simple interest?
Answers
Answer:
$4167
Step-by-step explanation:
Original Amount: $4167 (rounded)
First Year: $4167 + $416.7 = $4583.7
Second Year $4583.7 + $416.7 = 5000.4
what is the value of x^2 - 6x + 9 when x = 2 + i?
Answers
The Expression x^2 - 6x + 9 when x = 2 + i is -2i
To evaluate the expression x^2 - 6x + 9 when x = 2 + i, we substitute the value of x into the expression:
(2 + i)^2 - 6(2 + i) + 9
Simplifying the first term, we get:
(2 + i)^2 = 2^2 + 2(2)(i) + i^2 = 4 + 4i + i^2
Since i^2 = -1, we can substitute that in and simplify further:
(2 + i)^2 = 4 + 4i - 1 = 3 + 4i
Now we substitute this into the original expression:
(2 + i)^2 - 6(2 + i) + 9 = (3 + 4i) - 6(2 + i) + 9
Simplifying further, we get:
= 3 + 4i - 12 - 6i + 9
= 0 - 2i
= -2i
Therefore, the value of x^2 - 6x + 9 when x = 2 + i is -2i.
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[TRUE OR FALSE] whenever two variables are correlated, we assume that one variable is the cause of the observed effect from the other variable.
Answers
Whenever two variables are correlated, we assume that one variable is the cause of the observed effect from the other variable.
It is a false statement,
Two variables that move together, or in the same direction, are said to have a positive correlation. When one variable rises while the other rises or when one variable falls while the other falls, there is a positive correlation. These two separate variables are theoretically influenced by the same outside factors because they travel in the same direction.
Whenever two variables are correlated, we assume that one variable is the cause of the observed effect from the other variable.
It is a false statement,
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For Zendaya's lemonade recipe, 3 lemons are required to make 18 cups of lemonade. At what rate are lemons being used in cups of lemonade per lemon?
Answers
The unit rate of cups of lemonade per lemon is in the ratio of 6:1.
What is the unit rate?The unit rate is the ratio of two or more quantities or values that are compared to one another.
Unit rates are also known as the constants of proportionality, constant rate, rate of change, or slope.
Zendaya's Lemonade Recipe:The number of lemons required to make 18 cups of lemonade = 3 lemons
Rate of lemonade per lemon = 18/3 = 6/1
Ratio = 6:1 or 1:6
Conversely, the rate of lemon per cup of lemonade = 1/6
Thus, we can conclude that for Zendaya to make 6 cups of lemonade, she requires 1 lemon, or alternatively, the ratio of lemon to a cup of lemonade is 1:6.
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let f (x) = 4x-9. write a function g whose graph is a translation 1 unit right of the graph of f
Answers
Function g(x) = 4x -13 after the graph is translated 1 unit right of the graph of f(x) = 4x - 9.
As given in the question,
Function f(x) = 4x -9
Graph g(x) get translated 1 unit right of the graph of function f(x)
It means change in the value of x equals to x-1
g(x) = 4(x-1) -9
⇒ g(x) = 4x -4 -9
⇒ g(x) = 4x-13
Therefore, function g(x) = 4x -13 after the graph is translated 1 unit right of the graph of f(x) = 4x - 9.
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For breakfast, Mr. Hill bought a cup of coffee for $1.39 and a bagel for $1.85. What was his total cost?
Answers
Answer:
$3.24
Step-by-step explanation:
Cost for the coffee + Cost for the bagel
$1.39 + $1.85
$3.24
The area of a children square playground is 50m squared.
a. what is the exact length of the playground
2. pine logs 3m long are to be laid around the playground. How many logs will need to be bought?
Answers
Answer:
a. solution
Area = 50 m^2
Area of square = l^2
or, 50 m^2 = l^2
or, √50m^2 = l
or, 5√2 m= l
Hence, The exact length of the playground is 5√2 m.
A restaurant chef has designed a new set of dishes for his menu. His set of dishes contains 10 10 main courses, and he will select a subset of them to place on the menu each night. To ensure variety of main courses for his patrons, he wants to guarantee that a night's menu is neither completely contained in nor completely contains another night's menu. What is the largest number of menus he can plan using his 10 10 main courses subject to this requirement
Answers
Answer:
252 menu
Step-by-step explanation:
Total number courses, n = 10
The largest number of menu the chef can plan using his 10 main courses.
There are various combinations of menu obtainable from the available courses :
Ranging from :
nC0 to nCn
The largest number of menu he can plan is the highest output obtainable between :
10C0 to 10C10
10C0 = 1
10C1 = 10
10C2 = 45
10C3 = 120
10C4 = 210
10C5 = 252
10C6 = 210
10C7 = 120
10C8 = 45
10C9 = 10
10C10 = 1
Hence, the largest number of menu he can plan using his 10 main courses is 252
2) Lucas has saved $50. He earns
$18.00 per hour at his job. He plans to
buy ticket to go on a cruise for $2300.
Write the minimum number of hours
Lucas must work to buy the ticket.
Answers
Answer:
125 hours
Step-by-step explanation:
Lucas already has $50. Since he get $18 per hour, you should use that to get the answer. After a little while, I got 125×18 and got 2250. If you add up the $50 he already has, the minimum hours he has to work is 125 hours which is about 5 days.
Answer:
I think the answer is 125 hours
Step-by-step explanation:
To find the unit rate we divide $2300 divided by $18.00
$2300 ÷ 18 = 127.777778
We round that up to 128
We also need to find the unit rate of his saved money
$50 ÷ $18.00 = 2.77777777778
We round that up to 3
128 hours - 3 hours = 125 hours
Use the following statement to answer parts a) and b). One hundred raffle tickets are sold for $3 each. One prize of $500 is to be awarded. Winners do not have their ticket costs of $3 refunded to them. Raul purchases one ticket.
a) Determine his expected value.
b) Determine the fair price of a ticket.
Answers
Using the expected value of a discrete distribution, it is found that:
a) His expected value is of -$2.5.
b) The fair price of a ticket is of $0.5.
What is the mean of a discrete distribution?The expected value of a discrete distribution is given by the sum of each outcome multiplied by it's respective probability.
In this problem, the distribution for the net value of the ticket is:
P(X = 497) = 1/1000.P(X = -3) = 999/1000.Item a:
The expected value is given by:
\(E(X) = 497\frac{1}{1000} - 3\frac{999}{1000} = -2.5\)
His expected value is of -$2.5.
Item b:
With an unknown ticket price, the distribution is:
P(X = 500 - x) = 1/1000.P(X = -x) = 999/1000.The game is fair if E(X) = 0, hence:
\(\frac{500 - x}{1000} - \frac{999x}{1000} = 0\)
0.5 - x = 0
x = 0.
The fair price of a ticket is of $0.5.
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b. Passes through the point (2, -4) and is parallel to 3x + y = 5
c. Passes through the point (2, -4) and is perpendicular to 3x + y = 5
Answers
keeping in mind that parallel lines have exactly the same slope, let's check for the slope of the equation above
\(3x+y=5\implies y=\stackrel{\stackrel{m}{\downarrow }}{-3}x+5\qquad \impliedby \qquad \begin{array}{|c|ll} \cline{1-1} slope-intercept~form\\ \cline{1-1} \\ y=\underset{y-intercept}{\stackrel{slope\qquad }{\stackrel{\downarrow }{m}x+\underset{\uparrow }{b}}} \\\\ \cline{1-1} \end{array}\)
so we're really looking for the equation of a line that has a slope oif -3 and it passes through (2 , -4)
\((\stackrel{x_1}{2}~,~\stackrel{y_1}{-4})\hspace{10em} \stackrel{slope}{m} ~=~ - 3 \\\\\\ \begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{(-4)}=\stackrel{m}{- 3}(x-\stackrel{x_1}{2}) \implies y +4 = - 3 ( x -2) \\\\\\ y+4=-3x+6\implies {\Large \begin{array}{llll} y=-3x+2 \end{array}}\)
now, keeping in mind that perpendicular lines have negative reciprocal slopes
\(\stackrel{~\hspace{5em}\textit{perpendicular lines have \underline{negative reciprocal} slopes}~\hspace{5em}} {\stackrel{slope}{ -3 \implies \cfrac{-3}{1}} ~\hfill \stackrel{reciprocal}{\cfrac{1}{-3}} ~\hfill \stackrel{negative~reciprocal}{-\cfrac{1}{-3} \implies \cfrac{1}{ 3 }}}\)
so for this one, we're looking for the equation of a line whose slope is 1/3 and it passes through (2 , -4)
\((\stackrel{x_1}{2}~,~\stackrel{y_1}{-4})\hspace{10em} \stackrel{slope}{m} ~=~ \cfrac{1}{3} \\\\\\ \begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{(-4)}=\stackrel{m}{ \cfrac{1}{3}}(x-\stackrel{x_1}{2}) \implies y +4 = \cfrac{1}{3} ( x -2) \\\\\\ y+4=\cfrac{1}{3}x-\cfrac{2}{3}\implies y=\cfrac{1}{3}x-\cfrac{2}{3}-4\implies {\Large \begin{array}{llll} y=\cfrac{1}{3}x-\cfrac{14}{3} \end{array}}\)
whats is the spread of 4, 6, 7, 9, 9
Answers
So, the spread of the numbers 4, 6, 7, 9, 9 is 5.
What is spread?Spread can refer to different things depending on the context. Here are a few examples:
In finance, spread refers to the difference between the bid price and the ask price of a security or asset. The bid price is the highest price a buyer is willing to pay for a security, while the ask price is the lowest price a seller is willing to accept. The spread represents the cost of trading that security or asset and is typically measured in basis points (hundredths of a percentage point).
In sports betting, spread refers to the point spread or handicap that is assigned to a team in order to make the betting odds more even. For example, if a football team is favored to win by 7 points, they may be assigned a -7-point spread, meaning they would have to win by more than 7 points for a bet on them to be successful.
In epidemiology, spread refers to the transmission of a disease from one person to another. The spread of a disease can be influenced by a variety of factors, including the infectiousness of the disease, the proximity and frequency of contact between people, and the effectiveness of measures to prevent transmission (such as vaccines or masks).
In cooking, spread can refer to a variety of ingredients that are used to spread on bread or crackers, such as butter, jam, or peanut butter.
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