Let f,g∈F(R) be two functions bounded in R and let h(x)=(f∘g)(x)sinx+g(x)[1+sinx+cos(2x)],∀x∈R Show that h is bounded in R. Please explain any proofs/methodologies listed in the solution so I can better understand it. A detailed solution will result in a thumbs up. Listed below is an example you can use as well as a rubric for how the solution should look. Understand this is how the class is taught and how solutions are meant to look.

The equation shows |h(x)| ≤ C+3B for all x ∈ R, so h is bounded in R by the constant M = C+3B

To show that h is bounded in R, we need to show that there exists a constant M such that

|h(x)| ≤ M for all x ∈ R.

First, note that since f and g are bounded in R, there exist constants A, B, C, and D such that

|f(x)| ≤ A and |g(x)| ≤ B for all x ∈ R,

and

|f(g(x))| ≤ C for all x ∈ R.

As a rubric example,

Now consider the term

(f∘g)(x)sinx.

Since sinx is bounded between -1 and 1, we can write

|(f∘g)(x)sinx| ≤ C |sinx| ≤ C for all x ∈ R.

Similarly, we can write

|g(x)(1+sinx+cos(2x))| ≤ B(1+1+1) = 3B for all x ∈ R.

Therefore, we have

|h(x)| ≤ C+3B for all x ∈ R,

so h is bounded in R by the constant

M = C+3B.

Thus, we have shown that h is bounded in R.

For more such answers on Rubric

https://brainly.com/question/30386918

#SPJ11

We have |h(x)| ≤ M1 + 3M2 for all x in R. Let M = M1 + 3M2, then |h(x)| ≤ M for all x in R. Hence, h(x) is bounded in R.

To show that h(x) is bounded in R, we need to find a constant M such that |h(x)| ≤ M for all x in R.

First, note that since f and g are bounded in R, there exist constants M1 and M2 such that |f(x)| ≤ M1 and |g(x)| ≤ M2 for all x in R.

Then, we can bound each term in h(x) separately:

|f(g(x))sin(x)| ≤ M1|sin(x)| ≤ M1

|g(x)(1+sin(x)+cos(2x))| ≤ M2(1+|sin(x)|+|cos(2x)|) ≤ M2(1+1+1) = 3M2

Know more about constant here:

https://brainly.com/question/31730278

#SPJ11

A. [(-30)+(-5)]+[45÷(-5)]
B.[17-(-9)]+[(-39)÷(-3)]
C. [(-23)+(-9)+11]+[(-12)×4]
D.[17+(-40)]+[(-16)+12]
E.[23+(-19)]+[(-5)×8×(-3){
Repaso de números enteros y fraccionarios Chicos me ayudan a resolver esto plis con todo el procedimiento ​

anlatarak cozmicekseniz cevaplamayin:) ​

An 8th-grade class washed cars and pets to raise money for a children’s hospital. They charged $10 to wash a car and $5 to wash a pet. They raised $1,600 in one week, washing a total of 201 cars and pets. How much more money was raised washing cars than pets?
$1,399
$1,190
$780
$410

HELPPP!!!! 50 points​

Given the graph, what is in initial value of the function?

(5 points)

A coordinate grid is shown with x and y axes labeled from 0 to 7 at increments of 1. A straight line joins the ordered pair 0, 1 with the ordered pair 6, 7.


a
1

b
5

c
6

d
7

Write a polynomial in standard form that represents the area of the shaded region.

Pls help me with this problem

M is the midpoint of AB. If A ( 4,0) and M (2, 1.5), then find the coordinates of B

Solve for x.
5/7x + 1/7= 63

Somebody help me please

What is (f + g)(x)?
f(x) = -4x
g(x) = -x^2 + 6x

Anybody good in geometry?

WHATS THE ANSWER TO THIS PLS TIME IS RUNNING OUT

The measure of AOC is 90. Find the value of x.

Work out, giving your answer in its simplest form:
4/7 divided by 2 1/3

I need to know what the answer to this problem is the right answer 4/8

calculate the Unknown Missing angle measures ​

write a active sentence then rewrite it as a passive​

Area of a room is 500m square. The cupboard takes 6% of the room’s area
a:what is the area of the cupboard
b:The table is 1/10 of the room. Calculate the area of the table.
c: bed has an area of 110m square. What is the percentage of the room is the bed?

I need help with this​

Fill in the blanks to write the fraction 3/4 as in different ways

Lucas and Mitchell were asked to add the numbers 3.54 and 12.09.


Lucas stated that he completed the addition problem by using the following method:

First, I broke the numbers apart. I broke 3.54 into 3.00+0.50+0.04 and 12.09 into 12.00+0.09.

Then, I combined 0.04 and 0.09 to get 0.13.

Next, I combined 0.13 and 0.50 to get 0.63.

Then, I combined 3.00+12.00 to get 15.00.

Finally, I added the decimal to get 15.63.

Mitchell stated that he completed the addition problem by using the following method:

First, I added the whole numbers to get 3.00+12.00=15.00.

Next, I added the numbers in the tenths place, 0.50+0.00=0.50.

Then, I added the numbers in the hundredths place, 0.04+0.09=0.013.

I then added 0.50 and 0.013 to get 0.513.

Lastly, I added the decimal to the whole number to get 15.513.

Determine if Lucas and/or Mitchell had a flaw in their method. If they did, correct the flaw(s) so that both the methods used are correct. PLZ ANSWER THE QUESTION!

f (x) = 2x g(x) = 4x + 5 Find f (x) + 3g(x)

f(x)=1/2x show steps please!!

find f(-3)

is -2 9/11 a rational number or irrational number​

Is 3x-22=-3x+8 no solution?

perrys bikes rents bikes for $18 plus $4 per hour. Lea paid $42 to rent a bike for how many hours did she rent the bike?

Which of these division problems have the same quotient as 12.3 divided by 0.4? Check all that apply.

A. 123 ÷ 4
B. 1.23 ÷ 0.4
C. 1,230 ÷ 40
D. 123 ÷ 0.04

-12x = 3(y - 3)

y = ___

Use the relationship shown on this graph.