## reynad one year ago given f(x)=sinx and g(x)=x^2+1, what is f(g(x)) and g(f(x))?

1. jim_thompson5910

$\Large f(x) = \sin(x)$ $\Large f(\color{red}{x}) = \sin(\color{red}{x})$ $\Large f(\color{red}{g(x)}) = \sin(\color{red}{g(x)})$ Do you see how I replaced every x with g(x)?

that means you would end up with f(g(x))=sin^2x + 1 and using that, g(f(x)) = sin^2x+1 right?

3. jim_thompson5910

well we replace the g(x) on the right side with x^2 + 1

4. jim_thompson5910

so let me write out the full steps $\Large f(x) = \sin(x)$ $\Large f(\color{red}{x}) = \sin(\color{red}{x})$ $\Large f(\color{red}{g(x)}) = \sin(\color{red}{g(x)})$ $\Large f(\color{red}{g(x)}) = \sin\left(\color{red}{x^2+1}\right)$

5. jim_thompson5910

As for the other way around, it looks like you got it $\Large g(x) = x^2 + 1$ $\Large g(f(x)) = (f(x))^2 + 1$ $\Large g(f(x)) = (\sin(x))^2 + 1$ $\Large g(f(x)) = \sin^2(x) + 1$

6. jim_thompson5910

no you cannot distribute like that

7. jim_thompson5910

sin(x+1) does NOT turn into sin(x) + sin(1)

wait, so, sin(x^2 +1) cannot be simplified anymore?

9. jim_thompson5910

yeah it's as simplified as it gets