Question

decompose the function y=sin^2(ln(x+3)) into 4 basic functions f(x), g(x), h(x), and j(x) state the functions, and give the order of composition to get y

Answer 1

I'm writing the square in a different way, so it's easier to identify our functions.
\[\Large y=\left[\sin[\ln(x+3)]\right]^2\]It's still the same as what you have written though.


Let's work from the outside and move inward. :)
So what would be our `outermost function`?
\[\Large f(x)=(x)^2\]

Err hmm maybe this will work better if we move from the inside outward.. lemme think...
I'm just trying to think of which way will make more sense to you.

Answer 2

No maybe this will work.
So what we want to do now is: We'll stuff our next function `inside` of our outer function.
We want our next function to be,\[\Large g(x)=\sin(x)\]

So sticking g(x) into our function f gives us,\[\Large f\left[g(x)\right]=\left[\sin (x)\right]^2\]

Answer 3

Follow what I'm doing? :o
Composition of functions can be a little tricky.
So I'm just wondering how comfortable you are with this concept.

Answer 4

oh ok that makes sense! I just couldnt think of the fourth one so it would be
f(x)= sinx
j(x)=f(x)^2 ??? not sure how to write this one
g(x)=lnx
h(x)=x+3

Answer 5

Yah those functions look good!
Make sure you compose them in the correct order though.

With the way you labeled them, it looks like j(x) is the outermost function.

Answer 6

j(x)=x^2

Answer 7

ok so i write it inner to outer then?

Answer 8

just think of normal math

Answer 9

you are doing square of the sin of the ln of x + 3

Answer 10

or in more arbritrary terms

Answer 11

you are the doing the function1 of another function2 of another function3 of x+3

Answer 12

and x+3 can also be another function of x