I am trying to find the domain of composite function f(g(x). f(x)=x^2+1 and g(x)=square root of x-1.

Question

ganeshie8 · Answer

domain of a composite function is the intersection of domains of its parent funcitons

ganeshie8 · Answer

domain of f(g(x)) =   domain of f(x)   $\cap $   domain of g(x)

anonymous · Answer

What would happen if I plug in g(x) into f(x)?

ganeshie8 · Answer

plugin and see

ganeshie8 · Answer

$f( \color{red}{x} )=\color{red}{x}^2+1$

$f( \color{red}{g(x)} ) =  (\color{red}{g(x)})^2 + 1 $

anonymous · Answer

then it is square root of x-1 to the second power plus one, right?

ganeshie8 · Answer

yup !

anonymous · Answer

then, how do I find the domain?

ganeshie8 · Answer

$f( \color{red}{x} )=\color{red}{x}^2+1$

$f( \color{red}{g(x)} ) =  (\color{red}{g(x)})^2 + 1 $

$\qquad ~~=  (\sqrt{x-1})^2 + 1 $

ganeshie8 · Answer

since f(g(x)) is a composition of f(x) and g(x),
it needs to SATISFY BOTH the domains of f(x) and g(x)

ganeshie8 · Answer

can u find domain of f(x)
and g(x)

?

anonymous · Answer

Let's see .. the domain of f(x) is all real numbers and the domain of g(x) is x is greater of equal to one.

anonymous · Answer

I mean than greater of equal to one

ganeshie8 · Answer

yes, 

domain of f(x) :  all reals
domain of g(x) : $ x\ge 1$

ganeshie8 · Answer

so, the domain of $f(g(x))$ is   $x \ge 1$

cuz $x \ge 1$ is the intersection of both domains

ganeshie8 · Answer

another way to look at it is :  $x \ge 1$ satisfies both given functions' domains f(x) and g(x)

anonymous · Answer

ok thank you

ganeshie8 · Answer

u wlc :)

ganeshie8 · Answer

when u have time watch this video : https://www.youtube.com/watch?v=_zy7Uro7iCg

ganeshie8 · Answer

actually i was wrong earlier,
to find domain of f(g(x))
u need to find intersection of domains of g(x) and f(g(x))
however the answer is still same here : x >= 1 is the domain of f(g(x))