Answers
Step-by-step explanation:
f^-1 means the inverse function that finds the x responsible for the creation of a given y (functional result) value.
y = 3x - 1
3x = y + 1
x = 1/3(y + 1)
and now we rename x to y and y to x, so that we get a normal function definition :
y = 1/3(x + 1) = (1/3)x + 1/3
now we create the first derivative d/dx of this function :
y' = 1/3
since both f(x) and f^‐1(x) are line functions with a constant slope, it is not surprising that the first derivative (the function of the slope of the base function) is a constant function.
so, for any value of x (incl. the requested x = 11) d/dx f^-1(x) = 1/3.
therefore also d/dx f^-1(11) = 1/3.
y = 2×sqrt(x)
sqrt(x) = y/2
x = y²/4
again we rename the variables, so that we get a normal function definition for the inverse function :
y = x²/4
the first derivative is :
y' = 2x/4 = x/2
so, d/dx f^-1(6) = 6/2 = 3
Use the inverse function theorem: if f(x) is continuously differentiable at x = a, and given f(a) = b, then f is invertible around x = a, and
[tex]\left(f^{-1}\right)'(b) = \dfrac1{f'(a)}[/tex]
Both the given functions f(x) meet the conditions above. Note that you don't need to actually compute the inverse function itself.
If f(x) = 3x - 1, then f(x) = 11 when
3x - 1 = 11 ⇒ 3x = 12 ⇒ x = 4
Now if f(4) = 11, it follows from the inverse function theorem that
[tex]\left(f^{-1}\right)'(11) = \dfrac1{f'(4)} = \boxed{\dfrac13}[/tex]
since f'(x) = 3.
If f(x) = 2√x, then f(x) = 6 when
2√x = 6 ⇒ √x = 3 ⇒ x = ±9
But note that f(x) is defined only for x ≥ 0, so we must take x = 9. Now if f(9) = 6, the inverse function theorem says
[tex]\left(f^{-1}\right)'(6) = \dfrac1{f'(9)} = \dfrac1{\frac1{\sqrt9}} = \boxed{3}[/tex]
since f'(x) = 1/√x.
Related Questions
WHAT IS THE AREA OF THE CURVED SURFACE OF A CYLINDER OF BASE RADIUS 4 CM AND HEIGHT 10 CM
Answers
Answer:
The curved surface area of cylinder is 251.42 cm².
Step-by-step explanation:
Here's the required formula to find the curved surface area of cylinder :
[tex]\star\underline{\boxed{\sf{CSA_{(Cylinder)}} = 2\pi rh}}[/tex]
Csa = Curved surface areaπ = 22/7r = radius h = heightSubstituting all the given values in the formula to find the curved surface area of cylinder :
[tex]\implies{\tt{CSA_{(Cylinder)}= 2\pi rh}}[/tex]
[tex]{\implies{\tt{CSA_{(Cylinder)} = 2 \times \dfrac{22}{7} \times 4 \times 10}}}[/tex]
[tex]{\implies{\tt{CSA_{(Cylinder)} = 2 \times \dfrac{22}{7} \times 40}}}[/tex]
[tex]{\implies{\tt{CSA_{(Cylinder)} = \dfrac{2 \times 22 \times 40}{7}}}}[/tex]
[tex]{\implies{\tt{CSA_{(Cylinder)} = \dfrac{44 \times 40}{7}}}}[/tex]
[tex]{\implies{\tt{CSA_{(Cylinder)}= \dfrac{1760}{7}}}}[/tex]
[tex]{\implies{\tt{CSA_{(Cylinder)} \approx 251.42 }}}[/tex]
[tex]{\star{\underline{\boxed{\tt{\red{CSA_{(Cylinder)} \approx 251.42 \: {cm}^{2}}}}}}}[/tex]
Hence, the curved surface area of cylinder is 251.42 cm².
[tex]\rule{300}{2.5}[/tex]
Kane is training for a marathon. He starts by running 3 miles during every training session. Each week he plans to increase the distance of his run by 1/4 miles.
Let w be the number of weeks. Write an expression to show the distance Kane run in a training session after weeks.
PLEASE HURRY
in 6th grade explanation
Answers
Answer:
m = 0.25w + 3
Step-by-step explanation:
let m = miles
The basic fixed distance is 3 miles
And he adds 0.25 miles every week, for w weeks
So the equation is m = 0.25w + 3
-Chetan K
Which part of this problem should be simplified first?
(100 – 37 + 15) × 2 ÷ 3
Answers
Answer:
the parentheses should be the first.
Step-by-step explanation:
follow the steps using PEMDAS.
P- parentheses
E- exponent
M- multiply
D- divide
A- addition
S- subtraction
Answer:
The answer is 100 - 37. Do not put parenthesis when answering your question.
Step-by-step explanation:
Order the numbers 0.64, 2/3, 65%, and 7/10 from least to greatest.
(Please explain how you did it too-)
Answers
Answer:
the order is 0.64, 65%, 2/3, and 7/10
Step-by-step explanation:
because 2/3=0.66666, 65%=0.65, 7/10=0.7
Describe how to transform the graph of f(x) = ⎜x⎟ to obtain the graph of the related function g(x). Then draw the graph of g.
Answers
Answer:
ssss
Step-by-step explanation:
dddd
1/2x - 5 = 10 - 3/4x Please show work !
Answers
Answer:
x=[tex]\frac{1}{12}[/tex]
Step-by-step explanation:
1/2x - 5 = 10 - 3/4x
4x/2x - 5 × 4x = 10 × 4x - [tex]\frac{3 x 4x}{4x}[/tex]
2 -5 × 4x = 10 × 4 - 3
2 - 20x = 40x - 3
-20x - 40x = -3 - 2
-60x = -3 - 2
-60x = -5
x = [tex]\frac{-5}{-60}[/tex]
x = [tex]\frac{1}{12}[/tex]
Anna wants to bake $2\frac{1}{2}$ dozen cookies for a party. However, the recipe she has only makes a dozen cookies. If the original recipe calls for $1\frac{1}{4}$ cups of sugar, how many cups of sugar does she need to make the cookies for her party?
Answers
ANSWER: Loudest spliff
Like terms can be ________
this is fill - up question
Answers
Answer:
.the same variables and same powers
Step-by-step explanation:
In algebra, like terms are terms that have the same variables and powers. The coefficients do not need to match. Unlike terms are two or more terms that are not like terms, i.e. they do not have the same variables or powers.
HOPE IT HELPS...............
Write an expression to represent:
Nine less than the quotient of two and a number z.
Answers
Answer:
2 / z - 9
Step-by-step explanation:
Don't worry division always goes before subtraction so you don't need any parenthesis.
Tell me if I'm wrong :)
What is the solution for the inequality 4≤r/18+10?
Answers
[tex]4 \leq \dfrac{r}{18} +10\\\\\\\implies 4-10 \leq \dfrac{r}{18}\\\\\\\implies -6 \leq \dfrac r{18}\\\\\\\implies -6(18) \leq r\\\\\\\implies -108 \leq r\\\\\\\implies r \geq -108\\\\\\\text{Interval,}~ [-108 , \infty)[/tex]
Hayley learned a total os 13 appetizer recipes over the course of three weeks of culinary school. how many weeks does she need to complete to have learned 21 appetizers ?
Answers
Answer:
5 would be the answer.
Step-by-step explanation:
13 divided by 3(weeks) = 4.3
4.3 x 5 = 21.
= 5.
Brainliest)
There are 12000 fish in a lake. Because of pollution, the number of fish halves every year. What will be the population of fish after 5 years?
help
Answers
Answer:
375 fishes
Step-by-step explanation:
12000 divided by 2 is 6000 1st year
6000 divided by 2 is 3000 2nd year
3000 divided by 2 is 1500 3rd year
1500 divided by 2 is 750 4th year
750 divided by 2 is 375 5th year
(Score for Question 3:
of 3 points)
3. Describe the transformation that takes
(x)=x+1 to g(x) = -x +4.
Answer:
Answers
Answer:
It's a rotation of 90 degrees - doesn't matter which way - around the intersection of the two lines
Step-by-step explanation:
Claim: it's a 90 degree rotation around the intersection of f and g.
It's easy to spot the two lines are perpendicular. Let's assume the rotation doesn't happen at the point of intersection [tex]x_0[/tex], but slightly to the side, around let's say, [tex]x_0+\epsilon[/tex]; the new line will coincide with [tex]g(x)[/tex] if and only if [tex]\epsilon = 0[/tex], ie only the rotation happens around the intersection.
9. Which of the following relations are functions? Choose all that apply. Assume that each different variable
has a different value.
{(a,b),(0,0),(c.c),(e,d)}
{(c,e),(c,d),(0,b)}
{(6,5),(c,d),(d.c),(0,0)}
{(a,b),(b,c),(c,d),(d,e)}
Answers
Answer:
A and c sorry if I'm wrong
Triangle PIN is rotated -270 degrees about the origin.
Draw the image of this rotation.
Answers
If triangle PIN is rotated -270 degrees about the origin, the new point is at:
P'(-3, 2), I'(7, 7) and N'(7, -2)
Transformation is the movement of a point from its initial location to a new location. Types of transformation are translation, reflection, rotation and dilation.
If a point A(x, y) is rotated -270 degrees about the origin, the new point is at A'(-y, x).
The triangle PIN has vertices at P(2, 3), I(7, -7) and N(2, -7). If it is rotated -270 degrees about the origin, the new point is at:
P'(-3, 2), I'(7, 7) and N'(7, -2)
Find out more at: https://brainly.com/question/11707700
Answer:
P=(-3,2)
I=(7,7)
N=(7,2)
Step-by-step explanation:
khan
NEED HELP NOW! SOLVE THE INEQUALITY: -r/3 ≤ 6
The r/3 is not division.. its a fraction. thx
Also.. once one person has answered... please don't answer more... i don't want ore than 10 points taken away. thank you so much
Answers
Answer:
[tex]r\geq -18[/tex]
Step-by-step explanation:
[tex]\frac{-r}{3} \leq 6\\\rule{150}{0.5}\\(\frac{-r}{3})*3\leq 6*3\\\\-r \leq 18\\\\\frac{-r \leq 18}{-1}\\\\\boxed{ r\geq -18}[/tex]
Hope this helps!
Question in on picture attached
Answers
Answer: 12
Step-by-step explanation:
180 = 76 + (3x+3) + (6x-7) = 12
Solve for x
x/4=4/10
Give your answer as an improper fraction in its simplest form.
Answers
Answer:
1.6
hope this helps you!!!!!!!
Step-by-step explanation:
the answer which I have solved is in the image if you have any query then please ask...
what’s the system for 5x+21y=103 and -5x+23y=29
Answers
……………………………..
Question 19 of 25
f() = 379-2 and g(x) = 2x* 4. find (f- )(x).
A*2
C 3 - 2x - 6
D. 37 - 2x + 2
Answers
4. Find the mode of the set of data 12, 13, 18, 10, 13, 18, 12, 10, 18, 15. *
1 point
13
10
15
O
18
Answers
Answer:
18
Step-by-step explanation:
18 appeared the most
If the simple interest on 4,000 for 10 years is 2,400 then what is the interest rate
Answers
Answer:
6%
Step-by-step explanation:
I =Prt I being interest paid, P is Principal, r is rate, t is term
2400=4000x r x 10
2400=40000r
2400/40000=.06 convert to percentage by multiplying by 100 x. 06=6
please help for brainlist you need to be accurate tho
Answers
Answer:
3.5t +5
Step-by-step explanation:
We are looking for the total distance
5 miles is the first distance traveled.
Then after the break they travel at a rate of 3.5 miles per hour for t hours
That distance is 3.5t
The total distance traveled is
5+3.5t
Rewriting the order
3.5t +5
Miss Reyes earns $3500 per month from which income tax is deducted at 30%. Calculate her net pay.
Answers
Answer: 3395 dollars
Step-by-step explanation:
First, the question said that Miss Reyes earns 3500 dollars. But the tax is 30%. Than, the tax is 3500 times to 30%. If you calculate it, it will be 105 dollars. Now, we can subtract the 105 from 3500. It equals to 3395 dollars.
State what additional information is required in order to know that the triangles are congruent for the reason given. If possible, write the triangle congruence statement.
refer to attachment
Answers
Answer:
SU ≅ RPΔPQR ≅ ΔUTSStep-by-step explanation:
For SSS congruence, all three pairs of sides must be marked as congruent. That is, there needs to be 3 hash marks on sides SU and RP. (Note the order of the vertices.)
needed: SU ≅ RP
If we label the triangles in order by hash marks, the congruence statement becomes ...
ΔPQR ≅ ΔUTS
Solve for X
[tex]\frac{2x+9}{5} \leq \frac{7x-5}{6}[/tex]
Give answer as an improper fraction in its simplest form
Answers
Basically treat the inequality as an equal sign and do it. Note that to switch the sides, the inequality also has to be flipped.
please someone solve this please asap
Answers
Answer:
Step-by-step explanation:
12-b
13-c
14-a
15-d
16-b
17-c
18-d]
Step-by-step explanation:
wow,1st time u have asked important question
1) State the area of the given trapezoid.
5m
5m
8m
4m
A) 20 square meters
B) 26 square meters
C) 32 square meters
D) 40 square meters
Answers
Answer:
The answer would be 26
Step-by-step explanation:
First I divided the shape to make a rectangle and a right angle triangle, then i calculated the shape of the rectangle which came out to be 20. After that I calculated the area of the right angle triangle which is 6. So The answer would be 6 + 20 = 26
Help!! You have about 4+ hours to answer!! Help!!
Answers
Answer:
I think I'm doing this right?
Step-by-step explanation:
4.3
----
a. 4.3
b. 4.3
c. 4
Answer:
1. 4.30
2. 4.3
3. 4
Step-by-step explanation:
a. hundredths
4.30 is exactly in between 4.29 and 4.31
a. at the top: 4.31
a. at the bottom: 4.29
a. in the middle: 4.30
b. tenths
4.3 is exactly in between 4.2 and 4.4
b. at the top: 4.4
b. at the bottom: 4.2
b. in the middle: 4.3
c. ones
4 is in the middle of 3 and 5
c. at the top: 3
c. at the bottom: 5
c. in the middle: 4
Someone please help! Thank you in advance!
Answers
Answer:
58ft
Step-by-step explanation:
100-60=40
40+18=58
the answer is 58ft
(brainliest plz)