Composition of Functions and Sequence

1.2 I tried to sort it but obviously something is wrong...

syms x y 
f = 1/(x+1); h = exp(x); g = 1;

h = compose(f,g)

1.3 I have no idea whats it is about.

Answer 1

[rifat]

1.2 - Try to use finverse

1.3 - Use loops to solve

N=3; % number of terms
v=sqrt(2);
for i=1:N-1
v=sqrt(2+v);
end
v

To find convergence use a very large value of N

[Kondiik]

Thank you. I used this one for 1.2 with small modification.

N=3; % number of terms
v=sqrt(2);
v
for i=1:N-1
v=sqrt(2+v);
v
end

[Roger Stafford]

It is more elegant to use the equation given to you by Matt J:

 L=sqrt(2+L)

and solve for L. It is the value for which there is no change in the iteration and it is also the value your sequence converges to. You don't need to carry out the iteration for large N to find it, though if you do, you will see that these answers agree.

Answer 2

[Matt J]

1.3 To what value does this sequence converge

Hint - the formula for the recursion is

 a(n)=sqrt(2+a(n-1))

If a(n) converges to a limit L, then taking limits on both sides leads to the equation

 L=sqrt(2+L)

Answer 3

[Roger Stafford]

For problem 1.2 I strongly recommend reading up on the definition of function composition. For example:

 http://en.wikipedia.org/wiki/Function_composition

You are looking to define a function y = g(x) which has the property that

 f(g(x)) = f(y) = h(x) = exp(x)

where f(y) = 1/(y+1). It's easy to solve for y = g(x) in this equation. Then set x equal to 1.

[Kondiik]

How to set x = 1?

syms x y; % creat symbolic variables
f = (1/(x+1));
h = exp(x);
g = y;

x = 1;

y = solve(h == compose(f,g,x,y), y)

|y =

exp(-x) - x|