For the given functions f, g, and h, find fogoh and state the exact domain of fogoh. Please show all of your work.

Question

f(x)=lnx
g(x)=|x|-2
h(x)=4x-7

Answer 1

domains:

f: x>0
g: all reals
h: all reals

f(x) = lnx
f(g(h)) = ln(g(h))
= ln(|h|-2)
= ln(|4x-7|-2)

ln(u) is defined only for u>0
so, we need |4x-7|-2 > 0
|4x-7| > 2
so, if 4x-7>=0 (or x>=7/4), 4x-7 > 2 ==> x > 9/4

or, if 4x-7 < 0 (or x<7/4), -(4x-7) > 2 ==> x < 5/4

so, domain is (-oo,5/4) U [9/4,oo)