Given the functions f(x) = √x-8 and g(x) = 4/2x+1
a) Find (fog)(x) b) Find g^-1(f^-1(x))

Question

Given the functions f(x) = √x-8 and g(x) = 4/2x+1 
a) Find (fog)(x)        b) Find g^-1(f^-1(x))

anonymous · Answer

@ybarrap

ybarrap · Answer

try it and I'll check

ybarrap · Answer

be back in a few minutes after you try...

anonymous · Answer

$$fog(x)=\frac{ 4 }{2(\sqrt{x+8)+1} ? }$$

anonymous · Answer

second part I still don't understand if you could explain to me please how the inverse function will work

anonymous · Answer

@freckles

ybarrap · Answer

It might be easier if you write
$$
(f\circ g)(x) 	ext{ as } f(g(x))
$$
so
$$
f(g(x))=f\left (\cfrac{4}{2x}+1ight )=\sqrt{g(x)-8}=\sqrt{\left (\cfrac{ 4}{2x}+1ight )-8}
$$
I'll let you simplify.
Next, for the inverse compositions
Check that the following are the inverses of f and g:
$$
f^{-1}(x)=x^2+8\
g^{-1}(x)=\cfrac{2}{x-1}\
$$
Then
$$
\left (g^{-1}\circ f^{-1}ight )(x)=g^{-1}\left (f^{-1}(x)ight )\
=g^{-1}\left (x^2+8ight )=\cfrac{2}{f^{-1}(x)-1}=\cfrac{2}{(x^2+8)-1}
$$
Simplify and you're done!

Make sense?