# Thread: composite functions with sine

1. ## composite functions with sine

f(x) = sin (x)
g(x) = x^2 - x + 1

the question basically asks for a composite g o f, or g(f(x)). I get:

= sin (x^2 - x + 1)
= sin^2x - sin x + .... (i dont know how to multiply sin with 1, is it just sin(1) ?)

its kinda confusing for me, since sine isn't really a variable, it's more like a "modifier" for a variable

2. Originally Posted by snypeshow
= sin (x^2 - x + 1)
This is f(g(x)). You want g(f(x)).

= sin^2x - sin x + .... (i dont know how to multiply sin with 1, is it just sin(1) ?)
No, no, no and no. Never, EVER do that again. sin(x) is a function. It is not a modifier. Never, ever, split up its argument like that. Ever. "sin" means nothing - except in a religious sense. "sin" must have an argument.

Didn't we do g(f(x)) already in the Domain and Range question?

3. Originally Posted by TKHunny

Didn't we do g(f(x)) already in the Domain and Range question?
the domain and range question was multiplying 2 functions. this question is about composite functions.

Composite functions are when you have 2 functions (so f(x) and g(x)) and f o g means that g(x) replaces x in the function f(x)

so in the question, f(x) = sin x and g(x) = x^2 - x + 1
I'm looking for f(g(x)) (sorry, i said the opposite in my first post, my fault)

so f(g(x)) = sin (x^2 - x + 1)

how would i evaluate f(g(x))? or do i leave it as is?

(and yeah i guess im not too good with trig, as you could probably tell )

4. No worries. Just pin down what the notation means.

Originally Posted by snypeshow
so f(g(x)) = sin (x^2 - x + 1)
Good.

How would i evaluate f(g(x))? or do i leave it as is?
There is some push to "simplify". "Evaluate" really doesn't mean much.

5. Originally Posted by TKHunny
No worries. Just pin down what the notation means.

Good.

There is some push to "simplify". "Evaluate" really doesn't mean much.
so f(g(x)) = sin (x^2 - x + 1) can't be broken down any more right?

6. bump

and how would i find the domain and range for this question?

7. Originally Posted by snypeshow
so f(g(x)) = sin (x^2 - x + 1) can't be broken down any more right?

Using the trig identity

$\displaystyle \sin(a + b) = \sin a \cos b + \cos a \sin b$

you have

$\displaystyle \sin (x^2 - x + 1) = \sin ((x^2 - x) + 1) =$

$\displaystyle \sin (x^2 - x) \cos 1 + \cos(x^2 - x) \sin 1$

and you can break that down even further using

$\displaystyle \sin(a - b) = \sin a \cos b - \cos a \sin b$ and

$\displaystyle \cos(a - b) = \cos a \cos b + \sin a \sin b$

8. thanks!

I'm also looking for the domain and range for this function (specifically the range)

9. Originally Posted by snypeshow
thanks!

I'm also looking for the domain and range for this function (specifically the range)
What is the range of the sine function?