Find the domain of the composite function f ₒ g . Please show all of your work.

Question

f(x) = √x; g(x) = 2x + 12

Answer 1

f(x) = sqrt(x)
g(x) = 2x + 12
thus,
(fog)(x) = sqrt(2x + 12)
note that to get the domain, you must get the intersection of the 3 function domains.
for f(x) = sqrt(x), x cannot be less than zero thus the domain is [0 , +infinity)
for g(x) = 2x + 12, x can be any real number thus domain is (-infinity, +infinity).
for (fog)(x) = sqrt(2x + 12), the lowest possible value of the radicand (term inside the radical sign) is zero. thus equating 2x + 12 to zero,
2x + 12 = 0
x = -6
thus domain is [-6 , +infinity)

the intersection of the three domains is therefore,
[0, +infinity)

hope this helps~ :)